#  On the impact of spatial heterogeneity and drift rate in a three-patch two-
species Lotka-Volterra competition model over a stream††thanks: S. Chen is
supported by National Natural Science Foundation of China (Nos. 12171117,
11771109) and Shandong Provincial Natural Science Foundation of China (No.
ZR2020YQ01).

Shanshan Chen111Corresponding Author, Email<EMAIL_ADDRESS>
Department of Mathematics, Harbin Institute of Technology
Weihai, Shandong 264209, P. R. China
Jie Liu222Email<EMAIL_ADDRESS>
School of Mathematics, Harbin Institute of Technology
Harbin, Heilongjiang, 150001, P. R. China
Yixiang Wu333Email<EMAIL_ADDRESS>
Department of Mathematics, Middle Tennessee State University
Murfreesboro, Tennessee 37132, USA

###### Abstract

In this paper, we study a three-patch two-species Lotka-Volterra competition
patch model over a stream network. The individuals are subject to both random
and directed movements, and the two species are assumed to be identical except
for the movement rates. The environment is heterogeneous, and the carrying
capacity is lager in upstream locations. We treat one species as a resident
species and investigate whether the other species can invade or not. Our
results show that the spatial heterogeneity of environment and the magnitude
of the drift rates have a large impact on the competition outcomes of the
stream species.
Keywords: Lotka-Volterra competition model, patch environment, evolution of
dispersal, directed drift, random movement.
MSC 2020: 92D25, 92D40, 34C12, 34D23, 37C65.

##  1 Introduction

The species living in stream environment is subject both passive random
movement and directed drift [46]. Intuitively, the drift will carry
individuals to the downstream end, which may be crowded or hostile. However,
the random dispersal may drive the individuals to the upper stream locations,
which are usually more favorable for the species [23]. Therefore, the joint
impact of both undirectional and directed dispersal rates on the population
dynamics of the species are usually complicated and have attracted increasing
research interests recently [22, 25, 34, 39, 40, 41, 46].

Dispersal has profound effects on the distribution and abundance of organisms,
and understanding the mechanisms for the evolution of dispersal is a
fundamental question related to dispersal [26]. In the seminal works of
Hastings [19] and Dockery _et al._ [15], it has been shown that in a spatial
heterogeneous environment, when two competing species are identical except for
the random dispersal rate, evolution of dispersal favors the species with a
smaller dispersal rate. However, in an advective environment when individuals
are subject to both undirectional random dispersal and directed movement,
species with a faster dispersal rate can be selected [3, 4, 11].

Two species reaction-diffusion-advection competition models of the following
form has been proposed to study the evolution of dispersal for stream species
[28, 34, 35, 36, 37, 42, 47, 48, 50, 51, 52]:

$\begin{cases}u_{t}=d_{1}u_{xx}-q_{1}u_{x}+u[r(x)-u-v],&0<x<l,\;\;t>0,\\\
v_{t}=d_{2}v_{xx}-q_{2}v_{x}+v[r(x)-u-v],&0<x<l,\;\;t>0,\\\
d_{1}u_{x}(0,t)-q_{1}u(0,t)=d_{2}v_{x}(0,t)-q_{2}v(0,t)=0,&t>0,\\\
d_{1}u_{x}(l,t)-q_{1}u(l,t)=d_{2}v_{x}(l,t)-q_{2}v(l,t)=0,&t>0,\\\
u(x,0),v(x,0)\geq(\not\equiv)0,&0<x<l.\end{cases}$ (1.1)

In [47], the authors have studied the existence and properties of the positive
equilibrium of (1.1). In [28, 34, 48], the authors have treated species $u$ as
a resident species and studied the conditions under which the species $u$ only
semitrivial equilibrium is stable/unstable. Various results on the global
dynamics of (1.1) are presented in [36, 37, 42, 51, 52]. In particular, if
$r(x)$ is constant, the works [36, 37, 52] show that the species with a larger
diffusion rate and/or a smaller advection rate wins the competition. If $r(x)$
is a decreasing function, the authors in [51] use $q_{1}$ and $q_{2}$ as
bifurcation parameters to study the global dynamics of (1.1).

To study the evolution of dispersal in a river network, the authors in [23,
24] propose and investigate three-patch two-species Lotka-Volterra competition
models. Let $\bm{u}=(u_{1},u_{2},u_{3})$ and $\bm{v}=(v_{1},v_{2},v_{3})$ be
the population density of two competing species respectively, where $u_{i}$
and $v_{i}$ are the densities in patch $i$. Suppose that the dispersal
patterns of the individuals and the configuration of the patches are shown in
Fig. 1.

$1$$2$$3$$d+q$$d+q$$d$$d$ Figure 1: A stream with three patches, where $d$ is
the random movement rate and $q$ is the directed drift rate. Patch 1 is the
upstream end, and patch 3 is the downstream end.

The competition patch model over the stream network in Fig. 1 (with
$r_{1}=r_{2}=r_{3}$) in [23, 24] is:

$\begin{cases}\displaystyle\frac{du_{i}}{dt}=\displaystyle\sum_{j=1}^{3}(d_{1}D_{ij}+q_{1}Q_{ij})u_{j}+r_{i}u_{i}\left(1-\frac{u_{i}+v_{i}}{k_{i}}\right),&i=1,2,3,\;\;t>0,\\\
\displaystyle\frac{dv_{i}}{dt}=\displaystyle\sum_{j=1}^{3}(d_{2}D_{ij}+q_{2}Q_{ij})v_{j}+r_{i}v_{i}\left(1-\frac{u_{i}+v_{i}}{k_{i}}\right),&i=1,2,3,\;\;t>0,\\\
\bm{u}(0)=\bm{u}_{0}\geq(\not\equiv)\;\bm{0},\;\bm{v}(0)=\bm{v}_{0}\geq(\not\equiv)\;\bm{0},\end{cases}$
(1.2)

where $d_{1}$ and $d_{2}$ are random movement rates; $q_{1}$ and $q_{2}$ are
directed movement rates; $\bm{r}=(r_{1},r_{2},r_{3})$ is the growth rate;
$\bm{k}=(k_{1},k_{2},k_{3})$ is the carrying capacity; and two $3\times 3$
matrices $D=(D_{ij})$ and $Q=(Q_{ij})$ represent the random movement pattern
and directed drift pattern of individuals, respectively, where

$D=\begin{bmatrix}-1&1&0\\\ 1&-2&1\\\ 0&1&-1\\\
\end{bmatrix},\;\;Q=\begin{bmatrix}-1&0&0\\\ 1&-1&0\\\ 0&1&0\\\
\end{bmatrix}.$ (1.3)

We can write the model as

$\begin{cases}\displaystyle\frac{du_{1}}{dt}=-(d_{1}+q_{1})u_{1}+d_{1}u_{2}+r_{1}u_{1}\left(1-\displaystyle\frac{u_{1}+v_{1}}{k_{1}}\right),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\displaystyle\frac{du_{2}}{dt}=(d_{1}+q_{1})u_{1}-(2d_{1}+q_{1})u_{2}+d_{1}u_{3}+r_{2}u_{2}\left(1-\displaystyle\frac{u_{2}+v_{2}}{k_{2}}\right),\\\
\displaystyle\frac{du_{3}}{dt}=(d_{1}+q_{1})u_{2}-d_{1}u_{3}+r_{3}u_{3}\left(1-\displaystyle\frac{u_{3}+v_{3}}{k_{3}}\right),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\displaystyle\frac{dv_{1}}{dt}=-(d_{2}+q_{2})v_{1}+d_{2}v_{2}++r_{1}v_{1}\left(1-\displaystyle\frac{u_{1}+v_{1}}{k_{1}}\right),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\displaystyle\frac{dv_{2}}{dt}=(d_{2}+q_{2})v_{1}-(2d_{2}+q_{2})v_{2}+d_{2}v_{3}+r_{2}v_{2}\left(1-\displaystyle\frac{u_{2}+v_{2}}{k_{2}}\right),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\displaystyle\frac{dv_{3}}{dt}=(d_{2}+q_{2})v_{2}-d_{2}v_{3}+r_{3}v_{3}\left(1-\displaystyle\frac{u_{3}+v_{3}}{k_{3}}\right),\\\
\bm{u}(0)=\bm{u}_{0}\geq(\not\equiv)\;\bm{0},\;\bm{v}(0)=\bm{v}_{0}\geq(\not\equiv)\;\bm{0}.\end{cases}$
(1.4)

We assume $d_{1},d_{2},q_{1},q_{2}>0$ and $r_{i},k_{i}>0$ for $i=1,2,3$. We
adopt the same assumption in [23] on $\bm{k}=(k_{1},k_{2},k_{3})$:

  1. $(\bf{H})$

$k_{1}>k_{2}>k_{3}>0$.

Biologically, $(\bf{}{H})$ means that the upstream locations are more
favorable for both species. Model (1.4) has two semitrivial equilibria
$(\bm{u}^{*},\bm{0})$ and $(\bm{0},\bm{v}^{*})$.

Two-species Lotka-Volterra competition patch models have attracted many
research interests recently. Model (1.2) with $n$ patches in spatially
homogeneous environment (i.e. $r_{1}=\dots=r_{n}$ and $k_{1}=\dots=k_{n}$) has
been considered in our earlier papers [7, 8], but many techniques and results
there cannot be generalized to the situation when $\bm{k}$ is non-constant.
The authors in [18, 44] have studied the global dynamics of model (1.2) with
two patches and $q:=q_{1}=q_{2}$. They have showed that there exists a
critical drift rate such that below it the species with a smaller dispersal
rate wins the competition while above it the species with a larger dispersal
rate wins. In a competition model with two patches, the authors in [12, 17,
32] have showed that the species with more evenly distributed resources has
less competition advantage. In [9], the global dynamics of a Lotka-Volterra
competition patch model is classified under some assumptions on patches, which
requires $d_{1}/q_{1}=d_{2}/q_{2}$ in terms of (1.2). For more studies on
competition patch models, we refer to the works [2, 5, 6, 27, 30, 33, 43, 45,
49].

We will take an adaptive dynamics approach [14, 16] to analyze (1.4) by
viewing species $\bm{u}$ as the resident species and species $\bm{v}$ as the
mutant/invading species. We fix parameters $d_{1}$ and $q_{1}$ and vary
$d_{2}$ and $q_{2}$. We show that there exists a curve $q=q_{\bm{u}}^{*}(d)$
dividing the $(d_{2},q_{2})$-plane into two regions such that
$(\bm{u}^{*},\bm{0})$ is stable if and only if $(d_{2},q_{2})$ is above the
curve. Our results complement those in [23] by defining and analyzing the
curve $q=q_{\bm{u}}^{*}(d)$ and obtaining the global dynamics of model (1.4).
In particular, we show that if $q_{1}<\underline{q}$ the curve
$q=q_{\bm{u}}^{*}(d)$ is bounded (see Fig. 3) and if $q_{1}>\overline{q}$ it
is unbounded (see Fig. 4). This result is in sharp contrast with the
corresponding one for the model in spatially homogeneous environment
($k_{1}=k_{2}=k_{3}$) [7], where the curve $q=q_{\bm{u}}^{*}(d)$ is always
unbounded. We give explicitly parameter ranges for competitive exclusion and
conditions for coexistence/bistability in three cases ($q_{1}<\underline{q}$,
$\underline{q}\leq q_{1}\leq\overline{q}$ and $q_{1}>\overline{q}$). Our
results show that the magnitude of the drift rates and the spatial
heterogeneity of environment have an large impact on the competition outcomes
of the stream species.

Our paper is organized as follows. In section 2, we list some preliminary
results. In section 3, we state the main results on model (1.4). We give some
conclusive remarks and numerical simulations in section 4. The proofs of the
main results are presented in section 5. In the appendix, we show the
relations of $\underline{q}$, $\overline{q}$, and $q_{0}$. These relations are
implicitly included in the main results, and we prove them for reader’s
convenience.

##  2 Preliminary

Let $A=(a_{ij})_{n\times n}$ be a square matrix with real entries, $\sigma(A)$
be the set of all eigenvalues of $A$, and $s(A)$ be the spectral bound of $A$,
i.e. $s(A)=\max\\{{\rm Re}\lambda:\lambda\in\sigma(A)\\}$. The matrix $A$ is
called _irreducible_ if it cannot be placed into block upper triangular form
by simultaneous row and column permutations and _essentially nonnegative_ if
$a_{ij}\geq 0$ for all $1\leq i,j\leq n$ and $i\neq j$. By the Perron-
Frobenius Theorem, if $A$ is irreducible and essentially nonnegative, then
$s(A)$ is an eigenvalue of $A$ (called the _principal eigenvalue_ of $A$),
which is the unique eigenvalue associated with a nonnegative eigenvector. The
following result on the monotonicity of spectral bound can be found in [1,
10]:

######  Lemma 2.1.

Let $A=(a_{ij})_{n\times n}$ be an irreducible and essentially nonnegative
matrix and $M=\text{diag}(m_{i})$ be a real diagonal matrix. If $s(A)=0$, then

$\displaystyle\frac{d}{d\mu}s(\mu A+M)\leq 0$

for $\mu\in(0,\infty)$ and the inequality is strict except for the case
$m_{1}=\cdots=m_{n}$. Moreover,

$\lim_{\mu\rightarrow 0}s(\mu A+M)=\max_{1\leq i\leq
n}\\{m_{i}\\}\;\;\text{and}\;\;\lim_{\mu\rightarrow\infty}s(\mu
A+R)=\sum_{i=1}^{n}{\theta_{i}m_{i}},$

where $\theta_{i}\in(0,1)$, $1\leq i\leq n$, is determined by $A$ and
$\displaystyle\sum_{i=1}^{n}{\theta_{i}}=1$ (if $A$ has each column sum
equaling zero, then $\bm{\theta}=(\theta_{1},\dots,\theta_{n})^{T}$ is a
positive eigenvector of $A$ corresponding to eigenvalue $0$).

Let $\bm{m}=(m_{1},m_{2},m_{3})$ be a real vector. We write $\bm{m}\gg\bm{0}$
if $m_{i}>0$ for all $i=1,2,3$, and $\bm{m}>\bm{0}$ if $\bm{m}\geq\bm{0}$ and
$\bm{m}\neq\bm{0}$. Matrix $dD+qQ+\text{diag}(m_{i})$ is irreducible and
essentially nonnegative for any $d,q>0$, where $D$ and $Q$ are defined by
(1.3). By the Perron-Frobenius Theorem,
$s\left(dD+qQ+\text{diag}(m_{i})\right)$ is the principal eigenvalue of the
following eigenvalue problem:

$\displaystyle\sum_{j=1}^{3}(dD_{ij}+qQ_{ij})\phi_{j}+m_{i}\phi_{i}=\lambda\phi_{i},\;\;i=1,2,3.$
(2.1)

We need to consider the following single species patch model:

$\begin{cases}\displaystyle\frac{du_{i}}{dt}=\displaystyle\sum_{j=1}^{3}(dD_{ij}+qQ_{ij})u_{j}+r_{i}u_{i}\left(1-\frac{u_{i}}{k_{i}}\right),&i=1,2,3,\;\;t>0,\\\
\bm{u}(0)=\bm{u}_{0}>\bm{0}.\end{cases}$ (2.2)

The global dynamics of (2.2) is as follows:

######  Lemma 2.2.

Let $D$ and $Q$ be defined in (1.3), $\bm{r},\bm{k}\gg\bm{0}$, $d>0$, and
$q\geq 0$. Then model (2.2) admits a unique positive equilibrium
$\bm{u}^{*}\gg\bm{0}$, which is globally asymptotically stable.

###### Proof.

By [13, 31, 38], it suffices to show that $\bm{0}$ is unstable, i.e.

$s:=s\left(dD+qQ+\text{diag}(r_{i})\right)>0.$

Let $\phi^{T}=(\phi_{1},\phi_{2},\phi_{3})^{T}\gg\bm{0}$ with
$\sum_{i=1}^{3}\phi_{i}=1$ be the positive eigenvector of
$dD+qQ+\text{diag}(r_{i})$ corresponding to $s$. Multiplying $(1,1,1)$ to the
left of $dD\phi+qQ\phi+\text{diag}({r_{i}})\phi=s\phi$, we get
$s=\sum_{i=1}^{3}{{r_{i}\phi_{i}}}>0$. This proves the result. ∎

By Lemma 2.2, model (1.4) has two semitrivial equilibria $(\bm{u}^{*},\bm{0})$
and $(\bm{0},\bm{v}^{*})$, where $\bm{u}^{*}(\text{resp.,
}\bm{v}^{*})\gg\bm{0}$ is the positive equilibrium of (2.2) with $(d,q)$
replaced by $(d_{1},q_{1})$ (resp., $(d_{2},q_{2})$). Linearizing model (1.4)
at $(\bm{u}^{*},\bm{0})$, we can easily see that its stability is determined
by the sign of
$\lambda_{1}\left(d_{2},q_{2},\bm{1}-{\bm{u}^{*}}/{\bm{k}}\right)$, which is
the principal eigenvalue of the following eigenvalue problem:

$\begin{cases}\lambda\phi_{1}=-(d_{2}+q_{2})\phi_{1}+d_{2}\phi_{2}+r_{1}\left(1-\displaystyle\frac{u^{*}_{1}}{k_{1}}\right)\phi_{1},\\\
\lambda\phi_{2}=(d_{2}+q_{2})\phi_{1}-(2d_{2}+q_{2})\phi_{2}+d_{2}\phi_{3}+r_{2}\left(1-\displaystyle\frac{u^{*}_{2}}{k_{2}}\right)\phi_{2},\\\
\lambda\phi_{3}=(d_{2}+q_{2})\phi_{2}-d_{2}\phi_{3}+r_{3}\left(1-\displaystyle\frac{u^{*}_{3}}{k_{3}}\right)\phi_{3}.\end{cases}$
(2.3)

In particular, $(\bm{u}^{*},\bm{0})$ is locally asymptotically stable if
$\lambda_{1}\left(d_{2},q_{2},\bm{1}-{\bm{u}^{*}}/{\bm{k}}\right)<0$ and
unstable if
$\lambda_{1}\left(d_{2},q_{2},\bm{1}-{\bm{u}^{*}}/{\bm{k}}\right)>0$. Here, we
abuse the notation by denoting
$\bm{1}-{\bm{u}^{*}}/{\bm{k}}:=(1-u_{1}^{*}/k_{1},1-u_{2}^{*}/k_{2},1-u_{3}^{*}/k_{3})$.

##  3 Main results

We fix $d_{1},q_{1}>0,\bm{r},\bm{k}\gg\bm{0}$ and view species $\bm{u}$ as the
resident species and $\bm{v}$ as the invading/mutating species. We investigate
the dynamics of model (1.4) varying $(d_{2},q_{2})$. For this purpose, we
divide the first quadrant of the $(d,q)$-plane into six regions:

$\begin{split}G_{11}:=&\\{(d,q):\;d\geq
d_{1},q\geq\frac{q_{1}}{d_{1}}d,(d,q)\neq(d_{1},q_{1})\\},\\\
G_{12}:=&\\{(d,q):\;0<d<d_{1},q\geq q_{1}\\},\\\ G_{13}:=&\\{(d,q):\;d\geq
d_{1},q_{1}\leq q<\frac{q_{1}}{d_{1}}d,(d,q)\neq(d_{1},q_{1})\\},\\\
G_{21}:=&\\{(d,q):\;0<d\leq
d_{1},0<q\leq\displaystyle\frac{q_{1}}{d_{1}}d,(d,q)\neq(d_{1},q_{1})\\},\\\
G_{22}:=&\\{(d,q):\;d>d_{1},0<q\leq q_{1}\\},\\\ G_{23}:=&\\{(d,q):\;0<d\leq
d_{1},\frac{q_{1}}{d_{1}}d<q\leq q_{1},(d,q)\neq(d_{1},q_{1})\\}.\end{split}$
(3.1)

Figure 2: Illustration of the six regions of $(d,q)$-plane.

For readers’ convenience, we graph the six regions in Fig. 2.

###  3.1 Invasion curve

We consider the local stability of $(\bm{u}^{*},\bm{0})$ in this subsection.
Biologically, if $(\bm{u}^{*},\bm{0})$ is stable, then a small amount of
species $\bm{v}$ cannot invade species $\bm{u}$; if $(\bm{u}^{*},\bm{0})$ is
unstable, then a small amount of species $\bm{v}$ may be able to invade
species $\bm{u}$. We prove that there exists a curve $q=q_{\bm{u}}^{*}(d)$ in
the $(d,q)-$plane such that $(\bm{u}^{*},\bm{0})$ is locally asymptotically
stable if $(d_{2},q_{2})$ is above the curve and $(\bm{u}^{*},\bm{0})$ is
unstable if it is below the curve. To this end, we define

$d^{*}=\begin{cases}\infty,&\text{if}\;\;\sum_{i=1}^{3}r_{i}\left(1-\frac{u_{i}^{*}}{k_{i}}\right)\geq
0,\\\
d_{0},&\text{if}\;\;\sum_{i=1}^{3}r_{i}\left(1-\frac{u_{i}^{*}}{k_{i}}\right)<0,\end{cases}$
(3.2)

where $d=d_{0}>0$ is the unique root of
$\lambda_{1}(d,0,\bm{1}-\bm{u^{*}/k})=0$ if
$\sum_{i=1}^{3}r_{i}\left(1-{u_{i}^{*}}/{k_{i}}\right)<0$ (see the existence
of $d_{0}$ in Lemma 5.2). We have the following result about the local
stability/instability of the semitrivial equilibrium $(\bm{u}^{*},\bm{0})$:

######  Theorem 3.1.

Suppose that $(\bf{H})$ holds, $\bm{r}\gg\bm{0}$, and $d_{1},q_{1}>0$. Then
there exists a continuous function
$q=q_{\bm{u}}^{*}(d):(0,d^{*})\to\mathbb{R}_{+}$ passing through
$(d_{1},q_{1})$ such that the following statements hold for model (1.4):

  1. $\rm(i)$

If $(d_{2},q_{2})\in S_{1}$, then the semitrivial equilibrium
$(\bm{u}^{*},\bm{0})$ is locally asymptotically stable;

  2. $\rm(ii)$

If $(d_{2},q_{2})\in S_{2}$, then the semitrivial equilibrium
$(\bm{u}^{*},\bm{0})$ is unstable.

Here, $S_{1}\cup S_{2}$ is a partition of the first quadrant of the
$(d,q)$-plane defined as follows:

$\begin{split}S_{1}:=&\left\\{(d,q):\;0<d<d^{*},\;\;q>q_{\bm{u}}^{*}(d)\right\\}\cup
S_{1}^{*},\\\
S_{2}:=&\left\\{(d,q):\;0<d<d^{*},\;0<q<q_{\bm{u}}^{*}(d)\right\\},\end{split}$
(3.3)

where

$S^{*}_{1}=\begin{cases}\left\\{(d,q):d\geq
d^{*},\;\;q>0\right\\},&\;\;\text{if}\;\;d^{*}\neq\infty,\\\
\emptyset,&\;\;\text{if}\;\;d^{*}=\infty.\\\ \end{cases}$ (3.4)

######  Remark 3.2.

We call the curve in the first quadrant of $(d,q)$-plane defined by the
function $q=q_{\bm{u}}^{*}(d)$ in Theorem 3.1 the _invasion curve_. This curve
consists with all the points $(d,q_{\bm{u}}^{*}(d))$ such that
$\lambda_{1}(d,q_{\bm{u}}^{*}(d),\bm{1}-{\bm{u}^{*}}/{\bm{k}})=0$, i.e.,
$(\bm{u}^{*},\bm{0})$ is linearly neutrally stable. The invasion curve divides
the first quadrant into $S_{1}\cup S_{2}$, where $S_{1}$ is the region above
the curve and $S_{2}$ is the region below it. By Theorem 3.1,
$(\bm{u}^{*},\bm{0})$ is locally asymptotically stable in $S_{1}$ and unstable
in $S_{2}$.

In the following of this paper, we denote

$\begin{split}&\underline{q}:=\min\left\\{\displaystyle\frac{r_{1}}{k_{1}}(k_{1}-k_{2}),\displaystyle\frac{r_{3}}{k_{3}}(k_{2}-k_{3})\right\\},\\\
&\overline{q}:=\max\left\\{\displaystyle\frac{r_{1}}{k_{1}}(k_{1}-k_{2}),\displaystyle\frac{r_{3}}{k_{3}}(k_{2}-k_{3})\right\\}.\end{split}$
(3.5)

We take $\underline{q}$ and $\overline{q}$ as the threshold values for the
drift rates. Specifically, if a drift rate is below $\underline{q}$ (above
$\overline{q}$), we call it a _slow (large) drift_ ; if a drift rate is
between $\underline{q}$ and $\overline{q}$, we call it an _intermediate
drift_. These definitions coincide with those in [24] if $r_{1}=r_{2}=r_{3}$.
It turns out that the magnitude of drift rate $q_{1}$ will have a large impact
on the shape of the invasion curve and the dynamics of the model.

We have the following result about the invasion curve:

######  Proposition 3.3.

Suppose that $(\bf{H})$ holds, $\bm{r}\gg\bm{0}$, and $d_{1},q_{1}>0$. Let
$S_{1}$ and $S_{2}$ be defined in Theorem 3.1. Then the following statements
hold:

  1. ${\rm(i)}$

$G_{11}\subset S_{1}$ and $G_{21}\subset S_{2}$;

  2. ${\rm(ii)}$

If $q_{1}>\overline{q}$, then $G_{12}\subset S_{1}$ and $G_{22}\subset S_{2}$;

  3. ${\rm(iii)}$

If $q_{1}<\underline{q}$, then $G_{13}\subset S_{1}$ and $G_{23}\subset S_{2}$

We explore further properties of the invasion curve:

######  Proposition 3.4.

Suppose that $(\bf{H})$ holds, $\bm{r}\gg\bm{0}$, and $d_{1},q_{1}>0$. Let
$q=q_{\bm{u}}^{*}(d):(0,d^{*})\to\mathbb{R}_{+}$ be defined in Theorem 3.1.
Then the following statements hold:

  1. ${\rm(i)}$

$\lim_{d\to 0}q_{\bm{u}}^{*}(d)=q_{0}$, where

$q_{0}=\max\left\\{r_{1}\left(1-\frac{u_{1}^{*}}{k_{1}}\right),\
r_{2}\left(1-\frac{u_{2}^{*}}{k_{2}}\right)\right\\};$ (3.6)

  2. ${\rm(ii)}$

If $q_{1}<\underline{q}$, then

$d^{*}=d_{0}\;\;\text{and}\;\;\lim_{d\to d^{*}}q_{\bm{u}}^{*}(d)=0;$ (3.7)

  3. ${\rm(iii)}$

If $q_{1}>\overline{q}$, then

$d^{*}=\infty\;\;\text{and}\;\;\lim_{d\to\infty}\displaystyle\frac{q_{\bm{u}}^{*}(d)}{d}=\theta\;\;\text{for
some}\;\;\theta\in\left(0,\ \frac{q_{1}}{d_{1}}\right);$ (3.8)

  4. ${\rm(iv)}$

If $\underline{q}\leq q\leq\overline{q}$, then (3.7) holds when
$\sum_{i=1}^{3}r_{i}\left(1-{u_{i}^{*}}/{k_{i}}\right)<0$, and (3.8) holds
when $\sum_{i=1}^{3}r_{i}\left(1-{u_{i}^{*}}/{k_{i}}\right)\geq 0$.

######  Remark 3.5.

By Propositions 3.3-3.4, the invasion curve lies in $G_{12}\cup G_{22}$ when
the drift rate $q_{1}$ is small, and it lies in $G_{13}\cup G_{23}$ when
$q_{1}$ is large. Moreover, if $q_{1}$ is small, the invasion curve is defined
on a bounded interval $(0,d_{0})$; if $q_{1}$ is large, it is defined on
$(0,\infty)$ and has a slant asymptote $q=\theta d$ for some
$\theta\in(0,q_{1}/d_{1})$.

###  3.2 Competitive exclusion

In this subsection, we study the global dynamics of model (1.4) and find some
parameter ranges of $(d_{2},q_{2})$ such that competitive exclusion happens.
The relations of $\underline{q}$, $\overline{q}$ and $q_{0}$ are implicitly
included in the results below. However, for reader’s convenience, we include
the proof in the appendix.

Firstly, we consider the small drift case, i.e., $q_{1}<\underline{q}$.

######  Theorem 3.6.

Suppose that $(\bf{H})$ holds, $\bm{r}\gg\bm{0}$, and $d_{1},q_{1}>0$ with
$q_{1}<\underline{q}$. Then the following statements hold:

  1. ${\rm(i)}$

If $(d_{2},q_{2})\in G_{21}\cup G_{23}$, then the semi-trivial equilibrium
$(\bm{0},\bm{v}^{*})$ of (1.4) is globally asymptotically stable;

  2. ${\rm(ii)}$

If $(d_{2},q_{2})\in G_{11}\cup G_{12}^{*}\cup G_{13}$, then the semi-trivial
equilibrium $(\bm{u}^{*},\bm{0})$ of (1.4) is globally asymptotically stable.

Here, $G_{12}^{*}$ is defined by

$G^{*}_{12}=\\{(d_{2},q_{2}):(d_{2},q_{2})\in G_{12},q_{2}>\overline{q}\\}.$
(3.9)

Figure 3: Illustration of the result for the case $q_{1}<\underline{q}$. If
$(d_{2},q_{2})$ is above the curve $q=q_{\bm{u}}^{*}(d)$, then
$(\bm{u}^{*},\bm{0})$ is stable; and if $(d_{2},q_{2})$ is under the curve,
then $(\bm{u}^{*},\bm{0})$ is unstable. If $(d_{2},q_{2})\in G_{21}\cup
G_{23}$, $(\bm{0},\bm{v}^{*})$ is globally asymptotically stable; if
$(d_{2},q_{2})\in G_{11}\cup G_{12}^{*}\cup G_{13}$, $(\bm{u}^{*},\bm{0})$ is
globally asymptotically stable.

######  Remark 3.7.

Our results on model (1.4) for the small drift rate case are summarized in
Fig. 3. We have proved that competitive exclusion appears if $(d_{2},q_{2})$
falls into the blue and yellow regions of Fig. 3.

Next, we consider the large drift case, i.e., $q_{1}>\overline{q}$.

######  Theorem 3.8.

Suppose that $(\bf{H})$ holds, $\bm{r}\gg\bm{0}$, and $d_{1},q_{1}>0$ with
$q_{1}>\overline{q}$. Then the following statements hold:

  1. ${\rm(i)}$

If $(d_{2},q_{2})\in G_{21}\cup G_{22}\cup G_{23}^{*}$, then the semi-trivial
equilibrium $(\bm{0},\bm{v}^{*})$ is globally asymptotically stable;

  2. ${\rm(ii)}$

If $(d_{2},q_{2})\in G_{11}\cup G_{12}$, then the semi-trivial equilibrium
$(\bm{u}^{*},\bm{0})$ is globally asymptotically stable.

Here, $G_{23}^{*}$ is defined by

$G_{23}^{*}=\\{(d_{2},q_{2}):(d_{2},q_{2})\in G_{23},q_{2}<\underline{q}\\}.$
(3.10)

Figure 4: Illustration of the result for the case $q_{1}>\overline{q}$. If
$(d_{2},q_{2})$ is above the curve $q=q_{\bm{u}}^{*}(d)$, then
$(\bm{u}^{*},\bm{0})$ is stable; and if $(d_{2},q_{2})$ is under the curve,
then $(\bm{u}^{*},\bm{0})$ is unstable. If $(d_{2},q_{2})\in G_{21}\cup
G_{22}\cup G_{23}^{*}$, $(\bm{0},\bm{v}^{*})$ is globally asymptotically
stable; and if $(d_{2},q_{2})\in G_{11}\cup G_{12}$, $(\bm{u}^{*},\bm{0})$ is
globally asymptotically stable.

######  Remark 3.9.

Our results on model (1.4) for the large drift rate case are summarized in
Fig. 4. Different from the small drift rate case, the invasion curve is
unbounded. Again, we are able to prove that competitive exclusion happens if
$(d_{2},q_{2})$ falls into the blue and yellow regions of Fig. 4.

Then, we consider the intermediate drift case, i.e., $\underline{q}\leq
q_{1}\leq\overline{q}$, and we have the following result on the global
dynamics of model (1.4).

######  Theorem 3.10.

Suppose that $(\bf{H})$ holds, $\bm{r}\gg\bm{0}$, and $d_{1},q_{1}>0$ with
$\underline{q}\leq q_{1}\leq\overline{q}$. Let $G_{12}^{*}$ be defined by
(3.9) and $G_{23}^{*}$ be defined by (3.10). Then the following statements
hold:

  1. ${\rm(i)}$

If $(d_{2},q_{2})\in G_{21}\cup G_{23}^{*}$, then the semi-trivial equilibrium
$(\bm{0},\bm{v}^{*})$ is globally asymptotically stable;

  2. ${\rm(ii)}$

If $(d_{2},q_{2})\in G_{11}\cup G^{*}_{12}$, then the semi-trivial equilibrium
$(\bm{u}^{*},\bm{0})$ is globally asymptotically stable.

Figure 5: Illustration of the result for the case $\underline{q}\leq
q_{1}\leq\overline{q}$. If $(d_{2},q_{2})\in G_{21}\cup G_{23}^{*}$,
$(\bm{0},\bm{v}^{*})$ is globally asymptotically stable; and if
$(d_{2},q_{2})\in G_{11}\cup G^{*}_{12}$, $(\bm{u}^{*},\bm{0})$ is globally
asymptotically stable.

######  Remark 3.11.

Our results on model (1.4) for the intermediate drift rate case are summarized
in Fig. 5. In this case, the invasion curve may be defined on either a bounded
or an unbounded interval. However, we know that it must locate between the
yellow and blue regions in Fig. 5, where competitive exclusion happens.

In view of Theorems 3.6, 3.8, and 3.10, the global dynamics of model (1.4) in
$G_{11}\cup G_{21}$ is independent of $q_{1}$:

######  Corollary 3.12.

Suppose that $(\bf{H})$ holds, $\bm{r}\gg\bm{0}$, and $d_{1},q_{1}>0$. Then
the following statements hold:

  1. ${\rm(i)}$

If $(d_{2},q_{2})\in G_{11}$, then the semitrivial equilibrium
$(\bm{u}^{*},\bm{0})$ is globally asymptotically stable;

  2. ${\rm(ii)}$

If $(d_{2},q_{2})\in G_{21}$, then the semitrivial equilibrium
$(\bm{0},\bm{v}^{*})$ is globally asymptotically stable.

More importantly, we have the following result about the evolution of random
dispersal and directed drift rates.

######  Corollary 3.13.

Suppose that $(\bf{H})$ holds, $\bm{r}\gg\bm{0}$, and $d_{1},q_{1}>0$. Then
the following statements hold:

  1. ${\rm(i)}$

Fix $d_{1}=d_{2}$. If $q_{1}<q_{2}$, then the semitrivial equilibrium
$(\bm{u}^{*},\bm{0})$ is globally asymptotically stable; If $q_{1}>q_{2}$,
then the semitrivial equilibrium $(\bm{0},\bm{v}^{*})$ is globally
asymptotically stable;

  2. ${\rm(ii)}$

Fix $q_{1}=q_{2}<\underline{q}$. If $d_{1}<d_{2}$, then the semitrivial
equilibrium $(\bm{u}^{*},\bm{0})$ is globally asymptotically stable; If
$d_{1}>d_{2}$, then the semitrivial equilibrium $(\bm{0},\bm{v}^{*})$ is
globally asymptotically stable;

  3. ${\rm(ii)}$

Fix $q_{1}=q_{2}>\overline{q}$. If $d_{1}<d_{2}$, then the semitrivial
equilibrium $(\bm{0},\bm{v}^{*})$ is globally asymptotically stable; If
$d_{1}>d_{2}$, then the semitrivial equilibrium $(\bm{u}^{*},\bm{0})$ is
globally asymptotically stable.

######  Remark 3.14.

By Corollary 3.13, the species with a smaller drift rate always has
competitive advantage. If the drift rate is small, the species with smaller
random dispersal rate has competitive advantage; if the drift rate is large,
larger random dispersal rate is selected.

###  3.3 Coexistence and bistability

If $(d_{2},q_{2})$ is in the blank regions of Fig. 3-5, we show that
bistability and coexistence may occur. To this end, we explore the
stability/instability of the semi-trivial equilibrium
$(\bm{0},\bm{v}^{*}(d_{2},q_{2}))$ along the invasion curve, i.e.
$q_{2}=q_{u}^{*}(d_{2})$. Let

$\hat{\lambda}_{1}(d_{2}):=\lambda_{1}\left(d_{1},q_{1},\bm{1}-\frac{{\bm{v}^{*}}\left(d_{2},q_{\bm{u}}^{*}(d_{2})\right)}{\bm{k}}\right).$
(3.11)

Then $\hat{\lambda}_{1}(d_{1})=0$, the semi-trivial equilibrium
$(\bm{0},\bm{v}^{*}(d_{2},q^{*}_{\bm{u}}(d_{2})))$ is stable if
$\hat{\lambda}_{1}(d_{2})<0$, and
$(\bm{0},\bm{v}^{*}(d_{2},q^{*}_{\bm{u}}(d_{2})))$ is unstable if
$\hat{\lambda}_{1}(d_{2})>0$. The following result for the large drift case
can be proved similarly as [7, Theorem 5.4], so we omit the proof here.

######  Theorem 3.15.

Suppose that $(\bf{H})$ holds, $\bm{r}\gg\bm{0}$, and $d_{1},q_{1}>0$ with
$q_{1}>\overline{q}$. Let $q=q_{\bm{u}}^{*}(d):(0,d^{*})\to\mathbb{R}_{+}$ be
defined in Theorem 3.1 with $d^{*}=\infty$ (see Proposition 3.4 (iii)). Then
for any $d_{2}>0$, the following statements holds:

  1. $\rm(i)$

If $\hat{\lambda}_{1}(d_{2})<0$, then

$\hat{q}(d_{2}):=\inf\left\\{q>0:\ q>q_{\bm{u}}^{*}(d_{2})\ \text{and}\
\lambda_{1}\left(d_{1},q_{1},\bm{1}-\frac{{\bm{v}^{*}}\left(d_{2},q\right)}{\bm{k}}\right)\geq
0\right\\}$

exists and satisfies

$\begin{cases}\hat{q}(d_{2})\in\left(q_{\bm{u}}^{*}(d_{2}),q_{1}\right)\;\;&\text{for}\;\;d_{2}<d_{1},\\\
\hat{q}(d_{2})\in\left(q_{\bm{u}}^{*}(d_{2}),\displaystyle\frac{q_{1}}{d_{1}}d_{2}\right)\;\;&\text{for}\;\;d_{2}>d_{1}.\end{cases}$
(3.12)

Moreover, for any $q_{2}\in(q_{\bm{u}}^{*}(d_{2}),\hat{q}(d_{2}))$, both
semitrivial equilibria $(\bm{u}^{*},\bm{0})$ and $(\bm{0},\bm{v}^{*})$ are
locally asymptotically stable and model (1.4) admits an unstable positive
equilibrium.

  2. $\rm(ii)$

If $\hat{\lambda}_{1}(d_{2})>0$, then

$\hat{q}(d_{2}):=\sup\left\\{q>0:\ q<q_{\bm{u}}^{*}(d_{2})\ \text{and}\
\lambda_{1}\left(d_{1},q_{1},\bm{1}-\frac{{\bm{v}^{*}}\left(d_{2},q\right)}{\bm{k}}\right)\leq
0\right\\}$

exists and satisfies

$\begin{cases}\hat{q}(d_{2})\in\left(\displaystyle\frac{q_{1}}{d_{1}}d_{2},q_{\bm{u}}^{*}(d_{2})\right)\;\;&\text{for}\;\;d_{2}<d_{1},\\\
\hat{q}(d_{2})\in\left(q_{1},q_{\bm{u}}^{*}(d_{2})\right)\;\;&\text{for}\;\;d_{2}>d_{1}.\end{cases}$

Moreover, for any $q_{2}\in(\hat{q}(d_{2}),q_{\bm{u}}^{*}(d_{2}))$, both
semitrivial equilibria $(\bm{u}^{*},\bm{0})$ and $(\bm{0},\bm{v}^{*})$ are
unstable and model (1.4) admits a stable positive equilibrium.

######  Remark 3.16.

If $\underline{q}\leq q_{1}\leq\overline{q}$ (the intermediate drift case),
Theorem 3.15 (i)-(ii) holds for any $d_{2}<d_{1}$, and we omit the statement
to save space here.

The small drift rate case will be handled sightly different from the large
drift rate case. For any $\theta>0$, by Lemma 2.1 and Proposition 3.3 (ii),
the line $q=d\theta$ and the invasion curve $q=q_{\bm{u}}^{*}(d)$ have exactly
one intersection point $(d^{*}(\theta),d^{*}(\theta)\theta)$. So we can
reparameterize the invasion curve as follows:

$\begin{cases}d=d^{*}(\theta),\\\
q=q^{*}(\theta)=d^{*}(\theta)\theta,\end{cases}\;\;\theta>0.$ (3.13)

Let

$\tilde{\lambda}_{1}(\theta):=\lambda_{1}\left(d_{1},q_{1},\bm{1}-\frac{{\bm{v}^{*}}\left(d^{*}(\theta),q^{*}(\theta)\right)}{\bm{k}}\right),\;\;\theta>0.$
(3.14)

Then the semi-trivial equilibrium
$(\bm{0},\bm{v}^{*}\left(d^{*}(\theta),q^{*}(\theta)\right))$ is stable if
$\tilde{\lambda}_{1}(\theta)<0$ and unstable if
$\tilde{\lambda}_{1}(\theta)>0$. Noticing that $q_{\bm{u}}^{*}(d_{1})=q_{1}$,
we have $d^{*}(q_{1}/d_{1})=d_{1}$ and

$\tilde{\lambda}_{1}\left({q_{1}}/{d_{1}}\right)=\lambda_{1}\left(d_{1},q_{1},\bm{1}-\frac{{\bm{v}^{*}}\left(d_{1},q_{1}\right)}{\bm{k}}\right)=0.$

######  Theorem 3.17.

Suppose that $(\bf{H})$ holds, $\bm{r}\gg\bm{0}$, and $d_{1},q_{1}>0$ with
$0<q_{1}<\underline{q}$. Then for any $\theta>0$, the following statements
holds:

  1. $\rm(i)$

If $\tilde{\lambda}_{1}(\theta)<0$, then

$\tilde{d}^{*}(\theta):=\inf\left\\{d>0:\ d>d^{*}(\theta)\ \text{and}\
\lambda_{1}\left(d_{1},q_{1},\bm{1}-\frac{{\bm{v}^{*}}\left(d,d\theta\right)}{\bm{k}}\right)\geq
0\right\\}$

exists with $d^{*}(\theta)<\tilde{d}^{*}(\theta)$ such that for any
$(d_{2},q_{2})$ with $q_{2}=d_{2}\theta$ and
$d^{*}(\theta)<d_{2}<\tilde{d}^{*}(\theta)$ both semitrivial equilibria
$(\bm{u}^{*},\bm{0})$ and $(\bm{0},\bm{v}^{*})$ are locally asymptotically
stable and model (1.4) admits an unstable positive equilibrium.

  2. $\rm(ii)$

If $\tilde{\lambda}_{1}(\theta)>0$, then

$\tilde{d}^{*}(\theta):=\sup\left\\{d>0:\ d<d^{*}(\theta)\ \text{and}\
\lambda_{1}\left(d_{1},q_{1},\bm{1}-\frac{{\bm{v}^{*}}\left(d,d\theta\right)}{\bm{k}}\right)\leq
0\right\\}$

exists with $\tilde{d}^{*}(\theta)<d^{*}(\theta)$ such that for any
$(d_{2},q_{2})$ with $q_{2}=d_{2}\theta$ and
$\tilde{d}^{*}(\theta)<d_{2}<d^{*}(\theta)$ both semitrivial equilibria
$(\bm{u}^{*},\bm{0})$ and $(\bm{0},\bm{v}^{*})$ are unstable and model (1.4)
admits a stable positive equilibrium.

Moreover, $\tilde{d}^{*}(\theta)$ satisfies

$\begin{cases}(\tilde{d}^{*}(\theta),\tilde{d}^{*}(\theta)\theta)\in
G_{12}\;\;\text{if}\;\;\displaystyle\theta>\frac{q_{1}}{d_{1}},\\\
(\tilde{d}^{*}(\theta),\tilde{d}^{*}(\theta)\theta)\in
G_{22}\;\;\text{if}\;\;0<\theta<\displaystyle\frac{q_{1}}{d_{1}}.\\\
\end{cases}$ (3.15)

###### Proof.

We prove (i), and (ii) can be proved similarly. Fix $\theta>0$. Suppose
$\tilde{\lambda}_{1}(\theta)<0$. Let

$A=\left\\{d>0:\ d>d^{*}(\theta)\ \text{and}\
\lambda_{1}\left(d_{1},q_{1},\bm{1}-\frac{{\bm{v}^{*}}\left(d,d\theta\right)}{\bm{k}}\right)\geq
0\right\\}.$

By the proof of Theorem 3.6 (i), $(\bm{0},\bm{v}^{*})$ is unstable or
neutrally stable if $(d_{2},q_{2})\in G_{11}\cup G_{13}$, which yields
$A\neq\emptyset$. Since $\tilde{\lambda}_{1}(\theta)<0$, there exists
$\epsilon_{0}>0$ such that

$\lambda_{1}\left(d_{1},q_{1},\bm{1}-\frac{{\bm{v}^{*}}\left(d^{*}(\theta)+\epsilon,\theta(d^{*}(\theta)+\epsilon)\right)}{\bm{k}}\right)<0,\
\ \text{for any }0<\epsilon<\epsilon_{0}.$

Therefore, $\tilde{d}^{*}(\theta)$ exists with
$d^{*}(\theta)<\tilde{d}^{*}(\theta)$.

If $(d_{2},q_{2})$ satisfies $q_{2}=d_{2}\theta$ and
$\tilde{d}^{*}(\theta)<d_{2}<d^{*}(\theta)$, by the definition of
$\tilde{d}^{*}(\theta)$, we have

$\lambda_{1}\left(d_{1},q_{1},\bm{r-v^{*}}(d_{2},q_{2})\right)<0,$

which means that $(\bm{0},\bm{v}^{*})$ is locally asymptotically stable. By
Theorem 3.1, $(\bm{u}^{*},\bm{0})$ is also locally asymptotically stable. By
the monotone dynamical system theory [20, 21, 45], model (1.4) admits an
unstable positive equilibrium. Finally, it is easy to see that (3.15) holds by
Theorem 3.6. ∎

##  4 Discussions and numerical simulations

In this section, we discuss about the results of the paper and present some
numerical simulations.

Firstly, we address the impact of spatial heterogeneity on model (1.4). If the
environment is homogeneous, i.e. assumption (H) is replaced by
$k_{1}=k_{2}=k_{3}$, model (1.4) with $n$ patches has been investigated in our
recent paper [7]. The main results in [7] can be summarized by Fig. 6. In
particular, we prove that the invasion curve is between the lines $q=q_{1}$
and $q=q_{1}d/d_{1}$, $(\bm{u}^{*},\bm{0})$ is globally asymptotically stable
in $G_{1}$, and $(\bm{0},\bm{v}^{*})$ is globally asymptotically stable in
$G_{2}$. These results are independent of the magnitude of drift rate $q_{1}$
and are similar to the large drift rate case in this paper. Biologically, the
downstream end is crowded due to the drift and thereby less friendly compared
with the upstream end. If the environment perturbs from being uniformly
distributed and the upstream locations become advantageous, e.g. assumption
(H) holds, then a larger drift rate may compensate for it. This may explain
why the homogeneous environment case is similar to the larger drift case in
this paper.

Figure 6: Illustration of the main results for (1.4) with $k_{1}=k_{2}=k_{3}$
in [7]. The blue cure is the invasion curve, which always lies between the
lines $q=q_{1}$ and $q=q_{1}d/d_{1}$. Moreover, $(\bm{u}^{*},\bm{0})$ is
globally asymptotically stable if $(d_{2},q_{2})\in G_{1}$, and
$(\bm{0},\bm{v}^{*})$ is globally asymptotically stable if $(d_{2},q_{2})\in
G_{2}$.

Next, we consider the impact of drift rate on model (1.4). By Propositions 3.4
and 3.13, if the drift rate $q_{1}$ is small ($q_{1}<\underline{q}$), the
invasion curve $q=q^{*}_{\bm{u}}(d)$ is defined on a bounded interval and the
species with a smaller random dispersal rate is advantageous; if $q_{1}$ is
large ($q_{1}>\overline{q}$), the invasion curve is unbounded with a slant
asymptote $q=\theta d$ for some $\theta>0$ and larger random dispersal rate is
favored. The results for the small drift rate case align with the ones in the
seminal works [15, 19], which claim that the species with a smaller random
dispersal rate will always out-compete the other one in a spatial
heterogeneous environment, when both species randomly move in space and are
different only by the movement rate. When the drift rate becomes large, the
outcomes of the competition change dramatically, and the species with a larger
dispersal rate may win the competition.

Figure 7: The invasion curve for different values of $q_{1}$. Here,
$\bm{k}=(5,3,1)$, $\bm{r}=(1,2,1)$, and $d_{1}=1$. The threshold values for
the drift rates are $\underline{q}=0.4$ and $\overline{q}=2$.

We numerically explore the impact of the drift rate $q_{1}$ on the shape of
the invasion curve $q=q^{*}_{\bm{u}}(d)$. Fix $\bm{k}=(5,3,1)$,
$\bm{r}=(1,2,1)$, and $d_{1}=1$. Then we can compute the threshold values for
the drift rates: $\underline{q}=0.4$ and $\overline{q}=2$. In Fig. 7, we plot
the invasion curves for $q_{1}=0.2,0.5,1.2,4$. If $q_{1}=0.2$ or $0.5$, the
invasion curves seem to be bounded with $\partial_{d}q^{*}_{\bm{u}}(d_{1})<0$,
which indicates that a smaller random dispersal rate is favored when
$q_{1}=q_{2}$ and $d_{1}\approx d_{2}$. In sharp contrast with the smaller
drift rate cases, if $d_{1}=1.2$ or $4$, the invasion curves seem to be
unbounded with $\partial_{d}q^{*}_{\bm{u}}(d_{1})>0$. This simulation also
shows that the invasion curve maybe be bounded or unbounded for the
intermediate drift case ($\underline{q}<q_{1}<\overline{q}$).

Finally, we explore bistablility and coexistence phenomena of model (1.4). Let
$d_{1}=1,q_{1}=1.5$, $\bm{r}=(3,7,3)$, and $\bm{k}=(5,3,1)$. We graph the
invasion curve $(d^{*}(\theta),q^{*}(\theta))$ and $\tilde{\lambda}(\theta)$
in Fig. 8. The stability of $(\bm{0},\bm{v}^{*})$ when
$(d_{2},q_{2})=(d^{*}(\theta),q^{*}(\theta))$ is determined by the sign of
$\tilde{\lambda}(\theta)$. In Fig. 8, we can see that
$\tilde{\lambda}(\theta)$ changes sign, which means that both bistablility and
coexistence are possible. Indeed, if we choose $(d_{2},q_{2})=(3.088,1.239)$,
which is slightly below the invasion curve, then both $(\bm{u}^{*},\bm{0})$
and $(\bm{0},\bm{v}^{*})$ are locally asymptotically stable. As shown in Fig.
9, if the initial data is $(\bm{u}(0),\bm{v}(0))=((0.1,0.1,0.1),(5,5,5))$,
then the solution of (1.4) converges to $(\bm{0},\bm{v}^{*})$; if the initial
data is $(\bm{u}(0),\bm{v}(0))=((5,5,5),(0.1,0.1,0.1))$, then the solution
converges to $(\bm{u}^{*},\bm{0})$. Finally, we choose
$(d_{2},q_{2})=(10.28,0.03)$, which is slightly above the invasion curve
($(\bm{u}^{*},\bm{0})$ is unstable). Since $\tilde{\lambda}$ is positive,
$(\bm{0},\bm{v}^{*})$ is also unstable, and the model has at least one stable
positive equilibrium. We graph the solution of (1.4) for initial data
$(\bm{u}(0),\bm{v}(0))=((1,1,1),(1,1,1))$ and the solutions seem to converge
to a positive equilibrium.

Figure 8: (a) The invasion curve $(d^{*}(\theta),q^{*}(\theta))$ when
$d_{1}=1$, $q_{1}=1.5$, $\bm{r}=(3,7,3)$, and $\bm{k}=(5,3,1)$. (b) The sign
of the curve $\tilde{\lambda}_{1}(\theta)$ determines the stability of
$(\bm{0},\bm{v}^{*})$ when $(d_{2},q_{2})=(d^{*}(\theta),q^{*}(\theta))$.
Figure 9: Solutions of (1.4) with $d_{1}=1$, $q_{1}=1.5$, $d_{2}=3.088$,
$q_{2}=1.239$, $\bm{r}=(3,7,3)$, $\bm{k}=(5,3,1)$. For $(a)-(b)$, initial data
is $\bm{u}(0)=(0.1,0.1,0.1)$ and $\bm{v}(0)=(5,5,5)$, and species $\bm{v}$
wins the competition; for $(c)-(d)$, initial data is $\bm{u}(0)=(5,5,5)$ and
$\bm{v}(0)=(0.1,0.1,0.1)$, and species $\bm{u}$ wins the competition.
Figure 10: Solution of (1.4) with $d_{1}=1$, $q_{1}=1.5$, $d_{2}=10.28$,
$q_{2}=0.03$, $\bm{r}=(3,7,3)$, $\bm{k}=(5,3,1)$, and initial data is
$\bm{u}(0)=(5,5,5)$ and $\bm{v}(0)=(5,5,5)$. The two species seem to coexist.

##  5 Proofs for the invasion curve

In this section, we present the proofs of the results on the invasion curve
$q_{\bm{u}}^{*}(d)$. We begin with an analysis on $\bm{u}^{*}$. A similar
result of the following lemma when $r_{1}=r_{2}=r_{3}$ except for the sign of
$\sum_{i=1}^{3}r_{i}\left(1-{u^{*}_{i}}/{k_{i}}\right)$ can be found in [23].

######  Lemma 5.1.

Suppose that $(\bf{H})$ holds, $\bm{r}\gg\bm{0}$, $d_{1}>0$, and $q_{1}\geq
0$. Then the following statements on $\bm{u}^{*}$ hold:

  1. ${\rm(i)}$

$d_{1}u_{i+1}^{*}-(d_{1}+q_{1})u_{i}^{*}<0$ for $i=1,2$;

  2. ${\rm(ii)}$

$u^{*}_{1}<k_{1}$ and $u^{*}_{3}>k_{3}$;

  3. ${\rm(iii)}$

If $q_{1}>\overline{q}$, then $u^{*}_{1}<u^{*}_{2}<u^{*}_{3}$ and
$\sum_{i=1}^{3}r_{i}\left(1-\frac{u^{*}_{i}}{k_{i}}\right)>0$;

  4. ${\rm(iv)}$

If $q_{1}<\underline{q}$, then $u^{*}_{1}>u^{*}_{2}>u^{*}_{3}$ and
$\sum_{i=1}^{3}r_{i}\left(1-\frac{u^{*}_{i}}{k_{i}}\right)<0$.

###### Proof.

By (1.4), we have

$\begin{cases}d_{1}u^{*}_{2}-(d_{1}+q_{1})u^{*}_{1}=-r_{1}u^{*}_{1}\left(1-\displaystyle\frac{u^{*}_{1}}{k_{1}}\right),\\\
\left(d_{1}u_{3}^{*}-(d_{1}+q_{1})u^{*}_{2}\right)-\left(d_{1}u_{2}^{*}-(d_{1}+q_{1})u^{*}_{1}\right)=-r_{2}u^{*}_{2}\left(1-\displaystyle\frac{u^{*}_{2}}{k_{2}}\right),\\\
d_{1}u_{3}^{*}-(d_{1}+q_{1})u^{*}_{2}=r_{3}u^{*}_{3}\left(1-\displaystyle\frac{u^{*}_{3}}{k_{3}}\right).\\\
\end{cases}$ (5.1)

Suppose to the contrary that $d_{1}u^{*}_{2}-(d_{1}+q_{1})u^{*}_{1}\geq 0$.
Then by the first equation of (5.1), we have $u_{2}^{*}\geq u_{1}^{*}\geq
k_{1}$. This, together with assumption $(\bf{H})$ and the second equation of
(5.1), implies that $d_{1}u_{3}^{*}-(d_{1}+q_{1})u^{*}_{2}>0$ and
$u_{3}^{*}>k_{3}$, which contradicts the third equation of (5.1). Therefore,
we have $d_{1}u^{*}_{2}-(d_{1}+q_{1})u^{*}_{1}<0$. Similarly, we can prove
$d_{1}u_{3}^{*}-(d_{1}+q_{1})u^{*}_{2}<0$. This proves (i). By (i) and the
first and third equations of (5.1), we can easily obtain (ii).

The proof of (iv) is similar to that of (iii), so we only prove (iii) here.
Suppose $q_{1}>\overline{q}$. We rewrite (5.1) as follows:

$\begin{cases}d_{1}(u^{*}_{2}-u^{*}_{1})=-r_{1}u^{*}_{1}\left(1-\displaystyle\frac{q_{1}}{r_{1}}-\displaystyle\frac{u^{*}_{1}}{k_{1}}\right),\\\
d_{1}(u^{*}_{3}-u^{*}_{2})-(d_{1}+q_{1})(u^{*}_{2}-u^{*}_{1})=-r_{2}u^{*}_{2}\left(1-\displaystyle\frac{u^{*}_{2}}{k_{2}}\right),\\\
-(d_{1}+q_{1})(u^{*}_{3}-u^{*}_{2})=-r_{3}u^{*}_{3}\left(1+\displaystyle\frac{q_{1}}{r_{3}}-\displaystyle\frac{u^{*}_{3}}{k_{3}}\right).\\\
\end{cases}$ (5.2)

Suppose to the contrary that $u^{*}_{1}\geq u^{*}_{2}$. Then by the first
equation of (5.2), we have
$k_{1}-\displaystyle\frac{q_{1}k_{1}}{r_{1}}-u^{*}_{1}\geq 0$. Since
$q_{1}>\displaystyle\frac{r_{1}}{k_{1}}(k_{1}-k_{2})$, we obtain

$k_{2}>k_{1}-\displaystyle\frac{q_{1}k_{1}}{r_{1}}\geq u_{1}^{*}\geq
u_{2}^{*}.$

Then by the second equation of (5.2), we get $u^{*}_{2}>u^{*}_{3}$. This,
combined with $q_{1}>\displaystyle\frac{r_{3}}{k_{3}}(k_{2}-k_{3})$, yields

$k_{3}\left(1+\displaystyle\frac{q_{1}}{r_{3}}\right)>k_{2}>u^{*}_{2}>u^{*}_{3},$

which contradicts the last equation of (5.2). This proves
$u^{*}_{1}<u^{*}_{2}$.

Suppose to the contrary that $u^{*}_{2}\geq u^{*}_{3}$. Then by the last
equation of (5.2), we have $u_{3}^{*}\geq
k_{3}\left(1+\displaystyle\frac{q_{1}}{r_{3}}\right)$. By
$q_{1}>\overline{q}$, we obtain

$u_{2}^{*}\geq u_{3}^{*}\geq
k_{3}\left(1+\displaystyle\frac{q_{1}}{r_{3}}\right)>k_{2}>k_{1}-\frac{q_{1}k_{1}}{r_{1}}.$

Then by the second equation of (5.2), we have $u_{1}^{*}>u_{2}^{*}$. By the
first equation of (5.2), we get

$0>d_{1}(u^{*}_{2}-u^{*}_{1})=-r_{1}u^{*}_{1}\left(1-\displaystyle\frac{q_{1}}{r_{1}}-\displaystyle\frac{u^{*}_{1}}{k_{1}}\right)>0,$

which is a contradiction. This proves $u_{2}^{*}<u_{3}^{*}$.

Dividing the $i$-th equation of (5.2) by $u_{i}^{*}$, we have

$\begin{cases}-(d_{1}+q_{1})+d_{1}\frac{u^{*}_{2}}{u^{*}_{1}}+r_{1}\left(1-\frac{u_{1}^{*}}{k_{1}}\right)=0,\\\
(d_{1}+q_{1})\left(\frac{u^{*}_{1}}{u^{*}_{2}}-1\right)-d_{1}\left(1-\frac{u^{*}_{3}}{u^{*}_{2}}\right)+r_{2}\left(1-\frac{u_{2}^{*}}{k_{2}}\right)=0,\\\
(d_{1}+q_{1})\frac{u^{*}_{2}}{u^{*}_{3}}-d_{1}+r_{3}\left(1-\frac{u_{3}^{*}}{k_{3}}\right)=0.\end{cases}$
(5.3)

Adding up the equations in (5.3), we obtain

$\begin{split}&\sum_{i=2}^{3}\left((d_{1}+q_{1})\left(\frac{u_{i-1}^{*}}{u_{i}^{*}}-1\right)+d_{1}\left(\frac{u_{i}^{*}}{u_{i-1}^{*}}-1\right)\right)+\sum_{i=1}^{3}r_{i}\left(1-\frac{u_{i}^{*}}{k_{i}}\right)\\\
=&\sum_{i=2}^{3}\frac{(u_{i-1}^{*}-u_{i}^{*})((d_{1}+q_{1})u_{i-1}^{*}-d_{1}u_{i}^{*})}{u_{i-1}^{*}u_{i}^{*}}+\sum_{i=1}^{3}r_{i}\left(1-\frac{u^{*}_{i}}{k_{i}}\right)=0.\end{split}$
(5.4)

Then by (i) and $u_{1}^{*}<u_{2}^{*}<u_{3}^{*}$, we have
$\sum_{i=1}^{3}r_{i}\left(1-\frac{u^{*}_{i}}{k_{i}}\right)>0$. This proves
(iii). ∎

###  5.1 Proof of Theorem 3.1

We prove the existence of the invasion curve $q=q_{\bm{u}}^{*}(d)$ in this
subsection.

######  Lemma 5.2.

Suppose that $(\bf{H})$ holds, $\bm{r}\gg\bm{0}$, and $d_{1},q_{1}>0$. Then
the following statements hold about the semitrivial equilibrium
$(\bm{u}^{*},\bm{0})$ of (1.4):

  * ${\rm(i)}$

If $\sum_{i=1}^{3}r_{i}\left(1-{u_{i}^{*}}/{k_{i}}\right)\geq 0$, then for any
$d>0$ there exists $q_{\bm{u}}^{*}(d)>0$ such that
$\lambda_{1}\left(d,q_{\bm{u}}^{*}(d),\bm{1}-{\bm{u}^{*}}/{\bm{k}}\right)=0$,
$\lambda_{1}\left(d,q,\bm{1}-{\bm{u}^{*}}/{\bm{k}}\right)<0$ for all
$q>q_{\bm{u}}^{*}(d)$, and
$\lambda_{1}\left(d,q,\bm{1}-{\bm{u}^{*}}/{\bm{k}}\right)>0$ for all
$q<q_{\bm{u}}^{*}(d)$;

  * ${\rm(ii)}$

If $\sum_{i=1}^{3}r_{i}\left(1-{u_{i}^{*}}/{k_{i}}\right)<0$, then there
exists $d_{0}>0$ such that
$\lambda_{1}(d_{0},0,\bm{1}-{\bm{u}^{*}}/{\bm{k}})=0$,
$\lambda_{1}(d,0,\bm{1}-{\bm{u}^{*}}/{\bm{k}})<0$ for all $d>d_{0}$, and
$\lambda_{1}(d,0,\bm{1}-{\bm{u}^{*}}/{\bm{k}})>0$ for all $d<d_{0}$. Moreover,
the following results hold:

    1. ${\rm(ii_{1})}$

For any $d\in(0,d_{0})$, there exists $q_{\bm{u}}^{*}(d)>0$ such that the
statement in ${\rm(i)}$ holds;

    2. ${\rm(ii_{2})}$

For any $d\in[d_{0},\infty)$, we have
$\lambda_{1}(d,q,\bm{1}-{\bm{u}^{*}}/{\bm{k}})<0$ for all $q>0$.

###### Proof.

For simplicity, we denote
$\lambda_{1}(d,q):=\lambda_{1}\left(d,q,\bm{1}-{\bm{u}^{*}}/{\bm{k}}\right)$.
An essential step of the proof is to show the following claim.

_Claim 1:_ Fixing $d>0$, equation $\lambda_{1}(d,q)=0$ has at most one root
for $q\in[0,\infty)$.

_Proof of Claim:_ Let $\bm{\psi}$ be the positive eigenvector corresponding to
$\lambda_{1}(d,q)$ with $\sum_{i=1}^{3}\psi_{i}=1$. Then, we have

$\lambda_{1}(d,q)\psi_{i}=d\sum_{j=1}^{3}D_{ij}\psi_{j}+q\sum_{j=1}^{3}Q_{ij}\psi_{j}+r_{i}\left(1-\frac{u_{i}^{*}}{k_{i}}\right)\psi_{i},\;\;i=1,2,3.$
(5.5)

Differentiating (5.5) with respect to $q$ and denoting
${}^{\prime}=\partial/\partial q$, we obtain

$\lambda_{1}^{\prime}\psi_{i}+\lambda_{1}\psi^{\prime}_{i}=d\sum_{j=1}^{3}D_{ij}\psi^{\prime}_{j}+\sum_{j=1}^{3}Q_{ij}\psi_{j}+q\sum_{j=1}^{3}Q_{ij}\psi^{\prime}_{j}+r_{i}\left(1-\frac{u_{i}^{*}}{k_{i}}\right)\psi^{\prime}_{i},\;\;i=1,2,3.$
(5.6)

Multiplying (5.5) by $\psi_{i}^{\prime}$ and (5.6) by $\psi_{i}$ and taking
the difference of them, we have

$\lambda_{1}^{\prime}\psi^{2}_{i}=\sum_{j=1}^{3}(dD_{ij}+qQ_{ij})(\psi_{i}\psi^{\prime}_{j}-\psi_{i}^{\prime}\psi_{j})+\sum_{j=1}^{3}Q_{ij}\psi_{i}\psi_{j},\;\;i=1,2,3.$
(5.7)

Motivated by [7, Eq. (3.7)], we introduce
$(\beta_{1},\beta_{2},\beta_{3})=\left(1,{d}/(d+q),d^{2}/(d+q)^{2}\right)$.
Multiplying (5.7) by $\beta_{i}$ and summing up in $i$, we obtain

$\lambda_{1}^{\prime}\sum_{i=1}^{3}\beta_{i}\psi_{i}^{2}=\sum_{i=1}^{2}\left(\frac{d}{d+q}\right)^{i-1}\left(-\psi_{i}+\frac{d}{d+q}\psi_{i+1}\right)\psi_{i}.$
(5.8)

Suppose $\lambda_{1}\left(d,\tilde{q}\right)=0$ for some $\tilde{q}\geq 0$. By
Lemma 5.1 (ii), we see that

$\begin{split}&d\psi_{2}-(d+\tilde{q})\psi_{1}=-\psi_{1}r_{1}\left(1-\frac{u^{*}_{1}}{k_{1}}\right)<0,\\\
&d\psi_{3}-(d+\tilde{q})\psi_{2}=\psi_{3}r_{1}\left(1-\frac{u^{*}_{3}}{k_{3}}\right)<0.\end{split}$

Therefore by (5.8), we have $\lambda^{\prime}(d,\tilde{q})<0$. This proves the
claim.

According to the claim, whether the equation $\lambda_{1}(d,q)=0$ has a root
in $q$ is determined by the sign of $\lambda_{1}(d,0)$ and
$\lim_{q\to\infty}\lambda_{1}(d,q)$.

_Claim 2:_ $\lim_{q\to\infty}\lambda_{1}(d,q)<0$.

_Proof of claim:_ Adding up all the equations of (5.5), we have

$\lambda_{1}(d,q)=\sum_{i=1}^{3}r_{i}\left(1-\frac{u_{i}^{*}}{k_{i}}\right)\psi_{i},$
(5.9)

which implies that $\lambda_{1}(d,q)$ is bounded for $d,q>0$. So up to a
subsequence, we may assume $\lim_{q\to\infty}\bm{\psi}=\tilde{\bm{\psi}}$.
Dividing (5.5) by $q$ and taking $q\to\infty$, we obtain

$\sum_{i=1}^{3}Q_{ij}\tilde{\psi}_{j}=0,\;\;i=1,2,3,$

which yields $\tilde{\bm{\psi}}=(0,0,1)^{T}$. Then taking $q\to\infty$ in
(5.9), we have

$\lim_{q\to\infty}\lambda_{1}(d,q)=r_{3}\left(1-\frac{u_{3}^{*}}{k_{3}}\right)<0,$

where we have used Lemma 5.1 (ii) in the last step. This proves the claim.

By Lemma 2.1, $\lambda_{1}(d,0)$ is strictly decreasing in $d$ with

$\lim_{d\to 0}\lambda(d,0)=\max_{1\leq i\leq
3}r_{i}\left(1-\frac{u_{i}^{*}}{k_{i}}\right)>0\;\;\text{and}\;\;\lim_{d\to\infty}\lambda(d,0)=\displaystyle\frac{1}{3}\sum_{i=1}^{3}r_{i}\left(1-\frac{u_{i}^{*}}{k_{i}}\right),$

where we have used Lemma 5.1 (ii) again to see that $1-u_{1}^{*}/k_{1}>0$. Now
the desired results follow from this and Claims 1-2. ∎

We are ready to prove Theorem 3.1.
_Proof of Theorem 3.1._ Let $d_{0}$ be defined in Lemma 5.2, $d^{*}$ be
defined by (3.2), and $q=q^{*}_{\bm{u}}(d):(0,d^{*})\to\mathbb{R}_{+}$ be
defined in Lemma 5.2. Then Theorem 3.1 (i)-(ii) follows from Lemmas 5.1-5.2
and the fact that the stability/instability of $(\bm{u}^{*},\bm{0})$ is
determined by the sign of
$\lambda_{1}\left(d,q,\bm{1}-{\bm{u}^{*}}/{\bm{k}}\right)$. The continuity of
$q=q^{*}_{\bm{u}}(d)$ follows from
$\partial_{q}\lambda_{1}\left(d,q^{*}_{\bm{u}}(d),\bm{1}-{\bm{u}^{*}}/{\bm{k}}\right)<0$
(Claim 1 of Lemma 5.2) and the implicit function theorem. ∎

###  5.2 Proof of Propositions 3.3-3.4

First we prove the following useful result:

######  Lemma 5.3.

Suppose that $(\bf{H})$ holds, $\bm{r}\gg\bm{0}$, and $d_{1},q_{1}>0$. Then we
have

$\lambda_{1}\left(d_{2},q_{2},\bm{1}-\frac{\bm{u}^{*}}{\bm{k}}\right)\neq 0,$

if one of the following conditions holds:

  1. ${\rm(i)}$

$(d_{2},q_{2})\in G_{11}\cup G_{21}$;

  2. ${\rm(ii)}$

$q_{1}>\overline{q}$ and $(d_{2},q_{2})\in G_{12}\cup G_{22}$;

  3. ${\rm(iii)}$

$q_{1}<\underline{q}$ and $(d_{2},q_{2})\in G_{13}\cup G_{23}$.

###### Proof.

Suppose to the contrary that
$\lambda_{1}\left(d_{2},q_{2},\bm{1}-{\bm{u}^{*}}/{\bm{k}}\right)=0$, and let
$\bm{\phi}\gg 0$ be a corresponding eigenvector. Note that
$\lambda_{1}\left(d_{1},q_{1},\bm{1}-{\bm{u}^{*}}/{\bm{k}}\right)=0$ with a
corresponding eigenvector $\bm{u}^{*}$. Let

$\tilde{f}_{0}=\tilde{f}_{3}=0,\;\;\tilde{g}_{0}=\tilde{g}_{3}=0,$

and

$\tilde{f}_{j}=d_{1}u^{*}_{j+1}-(d_{1}+q_{1})u_{j}^{*},\;\;\tilde{g}_{j}=d_{2}\phi_{j+1}-(d_{2}+q_{2})\phi_{j},\;\;j=1,2.$
(5.10)

Then we have

$\displaystyle\tilde{f}_{j}-\tilde{f}_{j-1}=-r_{j}u^{*}_{j}\left(1-\displaystyle\frac{u_{j}^{*}}{k_{j}}\right),\;\;\;j=1,2,3,$
(5.11a)
$\displaystyle\tilde{g}_{j}-\tilde{g}_{j-1}=-r_{j}\phi_{j}\left(1-\displaystyle\frac{u_{j}^{*}}{k_{j}}\right),\;\;\;j=1,2,3.$
(5.11b)

Using similar arguments as in [7, Lemma 5.7], we can show

$\sum_{j=1}^{2}\left[(d_{1}-d_{2})(\phi_{j+1}-\phi_{j})-(q_{1}-q_{2})\phi_{j}\right]\tilde{f}_{j}{\displaystyle\frac{d_{1}^{j}}{\left(d_{1}+q_{1}\right)^{j+1}}}=0,$
(5.12)

and

$\sum_{k=1}^{2}\left[(d_{2}-d_{1})(u^{*}_{j+1}-u^{*}_{j})-(q_{2}-q_{1})u^{*}_{j}\right]\tilde{g}_{j}{\displaystyle\frac{d_{2}^{j}}{\left(d_{2}+q_{2}\right)^{j+1}}}=0.$
(5.13)

Indeed, multiplying (5.11a) by
$\left(\displaystyle\frac{d_{1}}{d_{1}+q_{1}}\right)^{j}\phi_{j}$, and summing
up from $j=1$ to $j=3$, we have

$\begin{split}&-\sum_{j=1}^{3}r_{j}u^{*}_{j}\phi_{j}\left(1-\displaystyle\frac{u_{j}^{*}}{k_{j}}\right)\left(\frac{d_{1}}{d_{1}+q_{1}}\right)^{j}\\\
=&\sum_{j=1}^{3}\left(\tilde{f}_{j}-\tilde{f}_{j-1}\right)\phi_{j}\left(\frac{d_{1}}{d_{1}+q_{1}}\right)^{j}\\\
=&\tilde{f}_{3}\phi_{3}\left(\frac{d_{1}}{d_{1}+q_{1}}\right)^{3}-\tilde{f}_{0}\phi_{1}\left(\frac{d_{1}}{d_{1}+q_{1}}\right)+\sum_{j=1}^{2}\tilde{f}_{j}\displaystyle\left(\frac{d_{1}}{d_{1}+q_{1}}\right)^{j}\left(\phi_{j}-\displaystyle\frac{d_{1}}{d_{1}+q_{1}}\phi_{j+1}\right)\\\
=&-\sum_{j=1}^{2}(d_{1}\phi_{j+1}-(d_{1}+q_{1})\phi_{j})\tilde{f}_{j}\displaystyle\frac{d_{1}^{j}}{\left(d_{1}+q_{1}\right)^{j+1}},\end{split}$
(5.14)

where we have used $\tilde{f}_{3}=\tilde{f}_{0}=0$ in the last step.
Similarly, multiplying (5.11b) by
$\left(\displaystyle\frac{d_{1}}{d_{1}+q_{1}}\right)^{j}u^{*}_{j}$ and summing
up from $j=1$ to $j=3$, we obtain

$\begin{split}-\sum_{j=1}^{3}r_{j}u^{*}_{j}\phi_{j}\left(1-\displaystyle\frac{u_{j}^{*}}{k_{j}}\right)\left(\frac{d_{1}}{d_{1}+q_{1}}\right)^{j}=-\sum_{j=1}^{2}\tilde{g}_{j}\tilde{f}_{j}\displaystyle\frac{d_{1}^{j}}{\left(d_{1}+q_{1}\right)^{j+1}}.\end{split}$
(5.15)

Taking the difference of (5.14) and (5.15), we obtain (5.12). Similarly, we
can prove (5.13).

By Lemma 5.1 (i)-(ii) and (5.11b), we have $\tilde{f}_{j},\tilde{g}_{j}<0$ for
$j=1,2$. Now we obtain a contradiction for each of (i)-(iii).

(i) We only consider the case $(d_{2},q_{2})\in G_{21}$, since the case
$(d_{2},q_{2})\in G_{11}$ can be studied similarly. Suppose $(d_{2},q_{2})\in
G_{21}$. Then we have $d_{2}\leq d_{1},q_{2}\leq q_{1}d_{2}/d_{1}$, and
$(d_{1},q_{1})\neq(d_{2},q_{2})$. If $d_{1}\neq d_{2}$, then it is easy to
check that

$\displaystyle\frac{q_{1}-q_{2}}{d_{1}-d_{2}}\geq\displaystyle\frac{q_{2}}{d_{2}}.$

(This inequality can be found in [52, Lemma 2.4].) This, together with
$\tilde{g}_{1},\tilde{g}_{2}<0$, yields

$(d_{1}-d_{2})(\phi_{j+1}-\phi_{j})-(q_{1}-q_{2})\phi_{j}<0,\;\;j=1,2.$ (5.16)

If $d_{1}=d_{2}$, then $q_{1}>q_{2}$ and (5.16) also holds. Then by
$\tilde{f}_{1},\tilde{f}_{2}<0$ and (5.12), we have

$0<\sum_{j=1}^{2}\left[(d_{1}-d_{2})(\phi_{j+1}-\phi_{j})-(q_{1}-q_{2})\phi_{j}\right]\tilde{f}_{j}{\displaystyle\frac{d_{1}^{j}}{\left(d_{1}+q_{1}\right)^{j+1}}}=0,$

which is a contradiction.

(ii) We only need to obtain a contradiction for the case $(d_{2},q_{2})\in
G_{22}$, since the case $(d_{2},q_{2})\in G_{12}$ can be studied similarly.
Suppose $(d_{2},q_{2})\in G_{22}$. Then we have $d_{2}>d_{1}$ and $q_{2}\leq
q_{1}$. By Lemma 5.1 (i), we have $u_{1}^{*}<u_{2}^{*}<u^{*}_{3}$, which
implies that

$(d_{2}-d_{1})(u^{*}_{j+1}-u^{*}_{j})-(q_{2}-q_{1})u^{*}_{j}>0,\ \ j=1,2.$

This, combined with $\tilde{g}_{1},\tilde{g}_{2}<0$ and (5.13), gives a
contradiction.

(iii) We only obtain a contradiction for the case $(d_{2},q_{2})\in G_{23}$,
since the case $(d_{2},q_{2})\in G_{13}$ can be studied similarly. Suppose
$(d_{2},q_{2})\in G_{23}$. Then we have $d_{2}\leq d_{1}$, $q_{2}\leq q_{1}$,
and $(d_{1},q_{1})\neq(d_{2},q_{2})$. By Lemma 5.1 (ii), we have
$u_{1}^{*}>u_{2}^{*}>u^{*}_{3}$, which implies that

$(d_{2}-d_{1})(u^{*}_{j+1}-u^{*}_{j})-(q_{2}-q_{1})u^{*}_{j}>0,\ \ j=1,2.$

This combined with $\tilde{g}_{1},\tilde{g}_{2}<0$ and (5.13) gives a
contradiction. ∎

We are ready to prove Propositions 3.3-3.4.

_Proof of Proposition 3.3._ (i) We only prove the case $G_{21}\subset S_{2}$,
(i.e., $(\bm{u}^{*},\bm{0})$ is unstable for $(d_{2},q_{2})\in G_{21}$), since
the case $G_{11}\subset S_{1}$ can be proved similarly. To avoid confusion, we
denote $\bm{u}^{*}$ by $\bm{u}^{*}_{\bm{k}}$. It is easy to see that
$\bm{u}^{*}_{\bm{k}}$ depends continuously on $\bm{k}$.

Suppose $(d_{2},q_{2})\in G_{21}$. We need to prove that
$\lambda_{1}\left(d_{2},q_{2},\bm{1}-{\bm{u}^{*}_{\bm{k}}}/{\bm{k}}\right)>0$.
Suppose to the contrary that
$\lambda_{1}\left(d_{2},q_{2},\bm{1}-{\bm{u}^{*}_{\bm{k}}}/{\bm{k}}\right)\leq
0$. By Lemma 5.3, we must have
$\lambda_{1}\left(d_{2},q_{2},\bm{1}-{\bm{u}^{*}_{\bm{k}}}/{\bm{k}}\right)<0$.
By [7, Theorem 4.2], we have
$\lambda_{1}\left(d_{2},q_{2},\bm{1}-{\bm{u}^{*}_{\bm{k}^{\prime}}}/{\bm{k}^{\prime}}\right)>0$,
where $\bm{k}^{\prime}=(k_{3},k_{3},k_{3})$.

Let
$\Lambda(s):=\lambda_{1}\left(d_{2},q_{2},\bm{1}-{\bm{u}^{*}_{\bm{k}_{1}(s)}}/{\bm{k}_{1}(s)}\right)$,
where $\bm{k}_{1}(s)=s\bm{k}+(1-s)\bm{k}^{\prime}$ satisfies (H) for any
$s\in[0,1]$. Since

$\Lambda(1)=\lambda_{1}\left(d_{2},q_{2},\bm{1}-\frac{\bm{u}^{*}_{\bm{k}}}{\bm{k}}\right)<0\;\;\text{
and}\;\;\Lambda(0)=\lambda_{1}\left(d_{2},q_{2},\bm{1}-\frac{\bm{u}^{*}_{\bm{k}^{\prime}}}{\bm{k}^{\prime}}\right)>0,$

there exists $s_{0}\in(0,1)$ such that $\Lambda(s_{0})=0$, which contradicts
Lemma 5.3.

(ii) By [7, Theorem 4.2], if $(d_{2},q_{2})\in G_{22}$ then
$\lambda_{1}\left(d_{2},q_{2},\bm{1}-{\bm{u}^{*}_{\bm{k}^{\prime}}}/{\bm{k}^{\prime}}\right)>0$;
and if $(d_{2},q_{2})\in G_{12}$, then
$\lambda_{1}\left(d_{2},q_{2},\bm{1}-{\bm{u}^{*}_{\bm{k}^{\prime}}}/{\bm{k}^{\prime}}\right)<0$,
where $\bm{k}^{\prime}=(k_{3},k_{3},k_{3})$. Then using similar arguments as
(i), we can prove (ii).

(iii) Let $\bm{\psi}$ be the positive eigenvector corresponding to
$\lambda_{1}:=\lambda_{1}\left(d_{2},q_{1},\bm{1}-{\bm{u}^{*}}/{\bm{k}}\right)$
with $\sum_{i=1}^{3}\psi_{i}=1$. Then, we have

$\lambda_{1}\psi_{i}=d_{2}\sum_{j=1}^{3}D_{ij}\psi_{j}+q_{1}\sum_{j=1}^{3}Q_{ij}\psi_{j}+r_{i}\left(1-\frac{u_{i}^{*}}{k_{i}}\right)\psi_{i},\;\;i=1,2,3.$
(5.17)

Differentiating (5.5) with respect to $d_{2}$ and denoting
${}^{\prime}=\partial/\partial d_{2}$, we obtain

$\lambda_{1}^{\prime}\psi_{i}+\lambda_{1}\psi^{\prime}_{i}=d_{2}\sum_{j=1}^{3}D_{ij}\psi^{\prime}_{j}+\sum_{j=1}^{3}D_{ij}\psi_{j}+q_{1}\sum_{j=1}^{3}Q_{ij}\psi^{\prime}_{j}+r_{i}\left(1-\frac{u_{i}^{*}}{k_{i}}\right)\psi^{\prime}_{i},\;\;i=1,2,3.$
(5.18)

Multiplying (5.17) by $\psi_{i}^{\prime}$ and (5.18) by $\psi_{i}$ and taking
the difference of them, we have

$\lambda_{1}^{\prime}\psi^{2}_{i}=\sum_{j=1}^{3}(d_{2}D_{ij}+q_{1}Q_{ij})(\psi_{i}\psi^{\prime}_{j}-\psi_{i}^{\prime}\psi_{j})+\sum_{j=1}^{3}D_{ij}\psi_{i}\psi_{j},\;\;i=1,2,3.$
(5.19)

Similar to the proof of Lemma 5.2, let
$(\beta_{1},\beta_{2},\beta_{3})=\left(1,{d_{2}}/(d_{2}+q_{1}),d_{2}^{2}/(d_{2}+q_{1})^{2}\right)$.
Multiplying (5.19) by $\beta_{i}$ and adding up them in $i$, we obtain

$\lambda_{1}^{\prime}\sum_{i=1}^{3}\beta_{i}\psi_{i}^{2}=\sum_{i=1}^{2}\left(\frac{d_{2}}{d_{2}+q_{1}}\right)^{i-1}\left(-\psi_{i}+\frac{d_{2}}{d_{2}+q_{1}}\psi_{i+1}\right)(\psi_{i}-\psi_{i+1}).$
(5.20)

Note that $\lambda_{1}\left(d_{1},q_{1},\bm{1}-{\bm{u}^{*}}/{\bm{k}}\right)=0$
with a corresponding eigenvector $(u_{1}^{*},u^{*}_{2},u_{3}^{*})^{T}$.
Moreover, by Lemma 5.1, we have $d_{1}u_{i+1}^{*}-(d_{1}+q_{1})u_{i}^{*}<0$
for $i=1,2$ and $u_{1}^{*}>u^{*}_{2}>u_{3}^{*}$. Then it follows from (5.20)
that

$\left.\frac{\partial\lambda_{1}\left(d_{2},q_{1},\bm{1}-{\bm{u}^{*}}/{\bm{k}}\right)}{\partial
d_{2}}\right|_{d_{2}=d_{1}}<0.$ (5.21)

This implies that
$\lambda_{1}\left(d_{2},q_{1},\bm{1}-{\bm{u}^{*}}/{\bm{k}}\right)>0$ if
$0<d_{1}-d_{2}\ll 1$ and
$\lambda_{1}\left(d_{2},q_{1},\bm{1}-\frac{\bm{u}^{*}}{\bm{k}}\right)<0$ if
$0<d_{2}-d_{1}\ll 1$. Then, by Lemma 5.3, we have $G_{13}\subset S_{1}$ and
$G_{23}\subset S_{2}$. ∎

######  Remark 5.4.

A similar inequality of (5.21) is proved in [23], and we include the proof for
completeness here.

_Proof of Proposition 3.4._ For any $0<d<d^{*}$, let $\bm{\psi}$ be the
eigenvector corresponding to
$\lambda_{1}\left(d,q_{\bm{u}}^{*}(d),\bm{1}-{\bm{u}^{*}}/{\bm{k}}\right)=0$
with $\bm{\psi}\gg 0$ and $\sum_{i=1}^{3}\psi_{i}=1$. Then

$d\sum_{j=1}^{3}D_{ij}\psi_{j}+q_{\bm{u}}^{*}(d)\sum_{j=1}^{3}Q_{ij}\psi_{j}+r_{i}\left(1-\frac{u_{i}^{*}}{k_{i}}\right)\psi_{i}=0,\;\;i=1,2,3.$
(5.22)

(i) Up to a subsequence, we may assume $\lim_{d\to
0}\bm{\psi}=\tilde{\bm{\psi}}$ for some $\tilde{\bm{\psi}}\geq\bm{0}$ and
$\sum_{i=1}^{3}\tilde{\psi}_{i}=1$. We first claim that $q_{\bm{u}}^{*}(d)$ is
bounded for $d\in(0,\delta)$ with $\delta\ll 1$. If it is not true, then
dividing (5.22) by $q_{\bm{u}}^{*}(d)$ and taking $d\to 0$, we have

$\sum_{j=1}^{3}Q_{ij}\tilde{\psi}_{j}=0,\;\;i=1,2,3,$ (5.23)

which yields $\tilde{\bm{\psi}}=(0,0,1)^{T}$. Adding up all the equations of
(5.22), we have

$\sum_{i=1}^{3}r_{i}\left(1-\frac{u_{i}^{*}}{k_{i}}\right)\tilde{\psi}_{i}=0.$
(5.24)

Taking $d\to 0$ in (5.24), we have $k_{3}-u_{3}^{*}=0$, which contradicts
Lemma 5.1 (ii). This proves the claim. By the claim, up to a subsequence, we
may assume $\lim_{d\to 0}q^{*}(\theta)=\tilde{q}_{0}\in[0,\infty)$.
Consequently, for sufficiently small $\epsilon>0$, there exists $\bar{d}>0$
such that $q_{\bm{u}}^{*}(d)<\tilde{q}_{0}+\epsilon$ for all $0<d<\bar{d}$. It
follows from Lemma 5.2 that

$\lambda_{1}\left(d,\tilde{q}_{0}+\epsilon,\bm{1}-\frac{\bm{u}^{*}}{\bm{k}}\right)<0$
(5.25)

for all $0<d<\bar{d}$. Hence,

$\begin{split}&\lim_{d\to
0}\lambda_{1}\left(d,\tilde{q}_{0}+\epsilon,\bm{1}-\frac{\bm{u}^{*}}{\bm{k}}\right)\\\
=&\displaystyle\max\left\\{r_{1}\left(1-\frac{u_{1}^{*}}{k_{1}}\right)-(\tilde{q}_{0}+\epsilon),r_{2}\left(1-\frac{u_{2}^{*}}{k_{2}}\right)-(\tilde{q}_{0}+\epsilon),r_{3}\left(1-\frac{u_{3}^{*}}{k_{3}}\right)\right\\}\leq
0.\\\ \end{split}$

Since $k_{3}-u_{3}^{*}=0$ and $\epsilon>0$ was arbitrary,

$\displaystyle\max\left\\{r_{1}\left(1-\frac{u_{1}^{*}}{k_{1}}\right)-\tilde{q}_{0},r_{2}\left(1-\frac{u_{2}^{*}}{k_{2}}\right)-\tilde{q}_{0}\right\\}\leq
0.$ (5.26)

Therefore, we have $\tilde{q}_{0}\geq q_{0}>0$. Similarly, we can prove
$\tilde{q}_{0}\leq q_{0}$. This proves (i).

Now we prove (ii)-(iv). If we show that (3.7) holds when
$\sum_{i=1}^{3}r_{i}\left(1-{u_{i}^{*}}/{k_{i}}\right)<0$, and (3.8) holds
when $\sum_{i=1}^{3}r_{i}\left(1-{u_{i}^{*}}/{k_{i}}\right)\geq 0$. Then (iv)
holds, and (ii)-(iii) follow from Theorem 5.1 (iii)-(iv).

By Lemma 5.2 (i), the function $q=q_{\bm{u}}^{*}(d)$ is defined for
$d\in(0,\infty)$ when
$\sum_{i=1}^{3}r_{i}\left(1-{u_{i}^{*}}/{k_{i}}\right)\geq 0$. We claim that
${q_{\bm{u}}^{*}(d)}/{d}$ is bounded for $d\in(\delta,\infty)$ for any fixed
$\delta>1$. If it is not true, up to a subsequence, we may assume

$\lim_{d\to\infty}\displaystyle\frac{q_{\bm{u}}^{*}(d)}{d}=\infty\;\;\text{and}\;\;\lim_{d\to\infty}\bm{\psi}=\hat{\bm{\psi}}$

for some $\hat{\bm{\psi}}\geq\bm{0}$ and $\sum_{i=1}^{3}\hat{\psi}_{i}=1$.
Then dividing (5.22) by $q_{\bm{u}}^{*}(d)$ and taking $d\to\infty$, we can
obtain a contradiction using similar arguments as in the proof of (i).
Therefore, ${q_{\bm{u}}^{*}(d)}/{d}$ is bounded for $d\in(\delta,\infty)$.
Then using similar arguments as in the proof of [7, Proposition 4.4], we can
show that (3.8) holds.

By Lemma 5.2 (ii), the function $q=q_{\bm{u}}^{*}(d)$ is defined for
$d\in(0,d_{0})$ when
$\sum_{i=1}^{3}r_{i}\left(1-{u_{i}^{*}}/{k_{i}}\right)<0$. Using similar
arguments as in (i), we can show that $q_{\bm{u}}^{*}(d)$ is bounded for
$d\in(d_{0}-\delta,d_{0})$ for some $\delta\ll 1$. Then, up to a subsequence,
we may assume

$\lim_{d\to d_{0}}q_{\bm{u}}^{*}(d)=\eta\;\;\text{and}\;\;\lim_{d\to
d_{0}}\bm{\psi}=\bm{\psi}^{*}$

for some ${\bm{\psi}^{*}}\geq\bm{0}$ and $\sum_{i=1}^{3}\psi^{*}_{i}=1$.
Taking $d\to d_{0}$ in (5.22), we see that

$d_{0}\sum_{j=1}^{3}D_{ij}\psi^{*}_{j}+\eta\sum_{j=1}^{3}Q_{ij}\psi^{*}_{j}+r_{i}\left(1-\frac{u_{i}^{*}}{k_{i}}\right)\psi^{*}_{i}=0,\;\;i=1,2,3,$
(5.27)

which yields
$\lambda_{1}\left(d_{0},\eta,\bm{1}-{\bm{u}^{*}}/{\bm{k}}\right)=0$. By the
proof of Lemma 5.2,
$\lambda_{1}\left(d_{0},q,\bm{1}-{\bm{u}^{*}}/{\bm{k}}\right)=0$ has at most
one root for $q\in[0,\infty)$. Since
$\lambda_{1}\left(d_{0},0,\bm{1}-{\bm{u}^{*}}/{\bm{k}}\right)=0$, we must have
$\eta=0$. This proves (3.7).∎

##  6 Proofs for the competitive exclusion results

Let $(\bm{u},\bm{v})$ be a positive equilibrium of model (1.4). Define

$\begin{split}&f_{0}=f_{3}=0,\;\;g_{0}=g_{3}=0,\\\
&f_{j}=d_{1}u_{j+1}-(d_{1}+q_{1})u_{j},\;\;g_{j}=d_{2}v_{j+1}-(d_{2}+q_{2})v_{j},\;\;j=1,2.\end{split}$
(6.1)

Clearly, we have

$f_{j}-f_{j-1}=-r_{j}u_{j}\left(1-\displaystyle\frac{u_{j}+v_{j}}{k_{j}}\right),\;\;\;j=1,2,3,$
(6.2)

and

$g_{j}-g_{j-1}=-r_{j}v_{j}\left(1-\displaystyle\frac{u_{j}+v_{j}}{k_{j}}\right),\;\;\;j=1,2,3.$
(6.3)

Then we have the following result about the sign of $f_{j},g_{j}$, $j=1,2$.

######  Lemma 6.1.

Suppose that $(\bf{H})$ holds, $\bm{r}\gg\bm{0}$, and
$d_{1},q_{1},d_{2},q_{2}>0$. If $(\bm{u},\bm{v})$ is a positive equilibrium of
model (1.4), then we have $f_{1},g_{1},f_{2},g_{2}<0$.

###### Proof.

First we prove $f_{1}<0$. Suppose to the contrary that $f_{1}\geq 0$. By
(6.2)-(6.3), we have $k_{1}-u_{1}-v_{1}\leq 0$ and $g_{1}\geq 0$. Since
$f_{1},g_{1}\geq 0$, we have $u_{2}>u_{1}$ and $v_{2}>v_{1}$. This combined
with $(\bf{H})$ implies that $k_{2}-u_{2}-v_{2}<0$. Then by (6.2)-(6.3) again,
we obtain that $f_{2},g_{2}>0$ and $k_{3}-u_{3}-v_{3}<0$, which contradicts
(6.2) with $j=3$. Therefore, we have $f_{1}<0$. Consequently, by (6.2)-(6.3)
with $j=1$, we have $g_{1}<0$. Using similar arguments, we can prove
$f_{2},g_{2}<0$. ∎

The following result is similar to [7, Lemma 5.7] with $j=1$ and $j=n=3$ (see
also the proof of Lemma 5.3). Thus, we omit the proof.

######  Lemma 6.2.

Suppose that $(\bf{H})$ holds, $\bm{r}\gg\bm{0}$, and
$d_{1},q_{1},d_{2},q_{2}>0$. If $(\bm{u},\bm{v})$ is a positive equilibrium of
model (1.4), then the following equations hold:

$\sum_{j=1}^{2}\left[(d_{1}-d_{2})(v_{j+1}-v_{j})-(q_{1}-q_{2})v_{j}\right]f_{j}{\displaystyle\frac{d_{1}^{j}}{\left(d_{1}+q_{1}\right)^{j+1}}}=0,$
(6.4)

and

$\sum_{j=1}^{2}\left[(d_{2}-d_{1})(u_{j+1}-u_{j})-(q_{2}-q_{1})u_{j}\right]g_{k}{\displaystyle\frac{d_{2}^{j}}{\left(d_{2}+q_{2}\right)^{j+1}}}=0.$
(6.5)

An essential step to prove the competitive exclusion results for model (1.4)
is to show the nonexistence of positive equilibrium:

######  Lemma 6.3.

Suppose that $(\bf{H})$ holds, $\bm{r}\gg\bm{0}$, and $d_{1},q_{1}>0$. Let
$G^{*}_{12}$ and $G_{23}^{*}$ be defined by (3.9) and (3.10), respectively.
Then model (1.4) admits no positive equilibrium, if one of the following
conditions holds:

  1. ${\rm(i)}$

$(d_{2},q_{2})\in G_{11}\cup G_{21}$;

  2. ${\rm(ii)}$

$q_{1}>\overline{q}$ and $(d_{2},q_{2})\in G_{12}\cup G_{22}\cup G_{23}^{*}$;

  3. ${\rm(iii)}$

$q_{1}<\underline{q}$ and $(d_{2},q_{2})\in G_{13}\cup G_{23}\cup G^{*}_{12}$.

  4. ${\rm(iv)}$

$\underline{q}\leq q_{1}\leq\overline{q}$ and $(d_{2},q_{2})\in G^{*}_{12}\cup
G_{23}^{*}$;

###### Proof.

Suppose to the contrary that model (1.4) admits a positive equilibrium
$(\bm{u},\bm{v})$. Then we will obtain a contradiction for each of the cases
(i)-(iv).

(i) We only consider the case $(d_{2},q_{2})\in G_{21}$. Since the nonlinear
terms of (1.4) are symmetric, the case $(d_{2},q_{2})\in G_{11}$ can be proved
similarly. Suppose $(d_{2},q_{2})\in G_{21}$. Then we have $d_{2}\leq
d_{1},q_{2}\leq q_{1}d_{2}/d_{1}$ and $(d_{1},q_{1})\neq(d_{2},q_{2})$. First,
we claim that

$(d_{1}-d_{2})(v_{j+1}-v_{j})-(q_{1}-q_{2})v_{j}<0\;\;\text{for}\;\;j=1,2.$
(6.6)

Indeed if $d_{1}=d_{2}$, then $q_{1}>q_{2}$ and (6.6) holds. If $d_{1}>d_{2}$,
then it is easy to check that

$\displaystyle\frac{q_{1}-q_{2}}{d_{1}-d_{2}}\geq\displaystyle\frac{q_{2}}{d_{2}}.$

(This inequality is in [52, Lemma 2.4]). This, combined with $g_{1},g_{2}<0$,
proves (6.6). Then by $f_{1},f_{2}<0$ and (6.4), we have

$0<\sum_{j=1}^{2}\left[(d_{1}-d_{2})(v_{j+1}-v_{j})-(q_{1}-q_{2})v_{j}\right]f_{j}{\displaystyle\frac{d_{1}^{j}}{\left(d_{1}+q_{1}\right)^{j+1}}}=0,$

which is a contradiction.

(ii) We first consider the case $(d_{2},q_{2})\in G_{22}$. Since the nonlinear
terms of (1.4) are symmetric, the case $(d_{2},q_{2})\in G_{12}$ can be proved
similarly. Suppose $(d_{2},q_{2})\in G_{22}$. Then, $d_{2}>d_{1}$ and
$q_{2}\leq q_{1}$. By (1.4), we have

$\displaystyle
d_{1}(u_{2}-u_{1})=-r_{1}u_{1}\left(1-\displaystyle\frac{q_{1}}{r_{1}}-\displaystyle\frac{u_{1}+v_{1}}{k_{1}}\right),$
(6.7a) $\displaystyle
d_{2}(v_{2}-v_{1})=-r_{1}v_{1}\left(1-\displaystyle\frac{q_{2}}{r_{2}}-\displaystyle\frac{u_{1}+v_{1}}{k_{1}}\right),$
(6.7b)
$\displaystyle(d_{1}+q_{1})(u_{1}-u_{2})-d_{1}(u_{2}-u_{3})=-r_{2}u_{2}\left(1-\displaystyle\frac{u_{2}+v_{2}}{k_{2}}\right),$
(6.7c)
$\displaystyle(d_{2}+q_{2})(v_{1}-v_{2})-d_{2}(v_{2}-v_{3})=-r_{2}v_{2}\left(1-\displaystyle\frac{u_{2}+v_{2}}{k_{2}}\right),$
(6.7d)
$\displaystyle(d_{1}+q_{1})(u_{2}-u_{3})=-r_{3}u_{3}\left(1+\displaystyle\frac{q_{1}}{r_{3}}-\displaystyle\frac{u_{3}+v_{3}}{k_{3}}\right),$
(6.7e)
$\displaystyle(d_{2}+q_{2})(v_{2}-v_{3})=-r_{3}u_{3}\left(1+\displaystyle\frac{q_{2}}{r_{3}}-\displaystyle\frac{u_{3}+v_{3}}{k_{3}}\right).$
(6.7f)

Then we show that $u_{1}<u_{2}<u_{3}$. Suppose to the contrary that $u_{1}\geq
u_{2}$. Then, by (6.7a), we see that

$u_{1}+v_{1}\leq k_{1}-\displaystyle\frac{q_{1}k_{1}}{r_{1}}\leq
k_{1}-\displaystyle\frac{q_{2}k_{1}}{r_{1}},$

where we have used $q_{2}\leq q_{1}$ in the last inequality. This, combined
with (6.7b), implies that $v_{1}\geq v_{2}$. Noticing that

$q_{1}>\overline{q}\geq\displaystyle\frac{r_{1}}{k_{1}}(k_{1}-k_{2}),$

we have

$u_{2}+v_{2}\leq u_{1}+v_{1}\leq
k_{1}-\displaystyle\frac{q_{1}k_{1}}{r_{1}}<k_{2},$

and consequently $u_{3}<u_{2}$ and $v_{3}<v_{2}$ by (6.7c)-(6.7d). This,
combined with $q_{1}>\overline{q}$, implies that

$u_{3}+v_{3}<u_{2}+v_{2}<k_{2}<k_{3}+\displaystyle\frac{k_{3}q_{1}}{r_{3}},$

which contradicts (6.7e). Similarly, we can show that $u_{2}<u_{3}$. So,
$u_{1}<u_{2}<u_{3}$, which leads to

$(d_{2}-d_{1})(u_{j+1}-u_{j})-(q_{2}-q_{1})u_{j}>0,\ \ j=1,2.$

Then by Lemma 6.1 and (6.5), we have

$0>\sum_{j=1}^{2}\left[(d_{2}-d_{1})(u_{j+1}-u_{j})-(q_{2}-q_{1})u_{j}\right]g_{j}{\displaystyle\frac{d_{2}^{j}}{\left(d_{2}+q_{2}\right)^{j+1}}}=0,$

which is a contradiction.

Now suppose that $(d_{2},q_{2})\in G_{23}^{*}$. Then $q_{2}<\underline{q}$ and
$(d_{1},q_{1})\in\hat{G}_{13}$, where

$\hat{G}_{13}:=\\{(d,q):\;d\geq d_{2},q_{2}\leq
q<\frac{q_{2}}{d_{2}}d,(d,q)\neq(d_{2},q_{2})\\}.$ (6.8)

Since the nonlinear terms of (1.4) are symmetric, this case can be proved
similarly as the case $(d_{2},q_{2})\in G_{13}$ (the proof is immediately
below).

(iii) Suppose that $(d_{2},q_{2})\in G_{13}$. Then we have $d_{2}\geq d_{1}$,
$q_{1}\leq q_{2}\leq q_{1}d_{2}/d_{1}$, and $(d_{1},q_{1})\neq(d_{2},q_{2})$.
We show that $u_{1}>u_{2}>u_{3}$. Suppose to the contrary that $u_{1}\leq
u_{2}$. Then, by (6.7a) and $q_{2}\geq q_{1}$, we have

$u_{1}+v_{1}\geq
k_{1}-\displaystyle\frac{q_{1}k_{1}}{r_{1}}>k_{1}-\displaystyle\frac{q_{2}k_{1}}{r_{1}}.$

This, combined with (6.7b), yields $v_{1}\leq v_{2}$. Noting that

$q_{1}<\underline{q}<\frac{r_{1}}{k_{1}}(k_{1}-k_{2}),$

we have

$k_{2}<k_{1}-\frac{q_{1}k_{1}}{r_{1}}\leq u_{1}+v_{1}\leq u_{2}+v_{2}.$

Then by (6.7c) and (6.7d), we have $u_{3}>u_{2}$ and $v_{3}>v_{2}$. Since
$q_{1}<\underline{q}$, we have

$k_{3}+\frac{k_{3}q_{1}}{r_{3}}<k_{2}<u_{2}+v_{2}<u_{3}+v_{3},$

which contradicts (6.7e). Similarly, we can show $u_{2}>u_{3}$. Therefore, we
have $u_{1}>u_{2}>u_{3}$ and

$(d_{2}-d_{1})(u_{j+1}-u_{j})-(q_{2}-q_{1})u_{j}<0,\;\;j=1,2.$

Then it follows from Lemma 6.1 and (6.5) that

$0<\sum_{j=1}^{2}\left[(d_{2}-d_{1})(u_{j+1}-u_{j})-(q_{2}-q_{1})u_{j}\right]g_{j}{\displaystyle\frac{d_{2}^{j}}{\left(d_{2}+q_{2}\right)^{j+1}}}=0,$

which is a contradiction.

For the case $(d_{2},q_{2})\in G_{23}$, using similar arguments as above, we
can obtain $v_{1}>v_{2}>v_{3}$, which leads to

$(d_{1}-d_{2})(v_{j+1}-v_{j})-(q_{1}-q_{2})v_{j}<0,\ \ j=1,2.$

This, combined with Lemma 6.1 and (6.4), implies that

$0<\sum_{j=1}^{2}\left[(d_{1}-d_{2})(v_{j+1}-v_{j})-(q_{1}-q_{2})v_{j}\right]f_{j}{\displaystyle\frac{d_{1}^{j}}{\left(d_{1}+q_{1}\right)^{j+1}}}=0,$

which is a contradiction.

Next suppose that $(d_{2},q_{2})\in G_{12}^{*}$. Then $q_{2}>\overline{q}$ and
$(d_{1},q_{1})\in\hat{G}_{22}$, where

$\hat{G}_{22}:=\\{(d,q):\;d>d_{2},0<q\leq q_{2}\\}.$

Since the nonlinear terms of (1.4) are symmetric, this case can be proved
similarly the case $(d_{2},q_{2})\in G_{22}$ in (ii).

(iv) If $(d_{2},q_{2})\in G_{23}^{*}$, the proof is the similar the
corresponding case in (ii); If $(d_{2},q_{2})\in G_{12}^{*}$, the proof is the
similar the corresponding case in (iii). ∎

We are ready to prove Theorems 3.6, 3.8 and 3.10.

_Proof of Theorem 3.6._ (i) Suppose that $q_{1}<\underline{q}$ and
$(d_{2},q_{2})\in G_{21}\cup G_{23}$. By Lemma 6.3 (i) and (iii), model (1.4)
admits no positive equilibrium. By Theorem 3.3 (i) and (iii),
$(\bm{u}^{*},\bm{0})$ is unstable. Then it follows from the monotone dynamical
system theory [20, 21, 29, 45] that $(\bm{u}^{*},\bm{0})$ is globally
asymptotically stable.

(ii) Suppose that $q_{1}<\underline{q}$ and $(d_{2},q_{2})\in G_{11}\cup
G_{12}^{*}$. By Lemma 6.3 (i) and (iii), model (1.4) admits no positive
equilibrium. By the monotone dynamical system theory [20, 21, 29, 45], it
suffices to show that $(\bm{0},\bm{v}^{*})$ is unstable. If $(d_{1},q_{1})\in
G_{12}^{*}$, then $q_{2}>\overline{q}$ and
$(d_{1},q_{1})\in\tilde{G}_{22}:=\\{(d,q):d>d_{2},0<q\leq q_{2}\\}$. Since the
nonlinear terms of model (1.4) are symmetric, it follows from Proposition 3.3
(ii) that $(\bm{0},\bm{v}^{*})$ is unstable. If $(d_{2},q_{2})\in G_{11}$,
then $(d_{1},q_{1})\in\tilde{G}_{21}$, where

$\tilde{G}_{21}:=\left\\{(d,q):0<d\leq
d_{2},0<q\leq\displaystyle\frac{q_{2}}{d_{2}}d,(d,q)\neq(d_{2},q_{2})\right\\}.$

Similarly, it follows from Proposition 3.3 (i) that $(\bm{0},\bm{v}^{*})$ is
unstable.

Finally, suppose that $q_{1}<\underline{q}$ and $(d_{2},q_{2})\in G_{13}$. By
Proposition 3.3 (iii), $(\bm{u}^{*},\bm{0})$ is locally asymptotically stable.
By Lemma 6.3 (iv), model (1.4) admits no positive equilibrium. If
$(\bm{0},\bm{v}^{*})$ is locally asymptotically stable, then model (1.4)
admits one unstable positive steady state, which is a contradiction. If
$(\bm{0},\bm{v}^{*})$ is unstable, then the monotone dynamical system theory
[20, 21, 29, 45] implies that $(\bm{u}^{*},\bm{0})$ is globally asymptotically
stable. If $(\bm{0},\bm{v}^{*})$ is neutrally stable, by [29, Theorem 1.4],
$(\bm{u}^{*},\bm{0})$ is globally asymptotically stable. This proves (ii). ∎

_Proof of Theorems 3.8 and 3.10._ We only need to prove the case
$(d_{2},q_{2})\in G_{23}^{*}$, since the other cases can be proved using
similar arguments in the proof of Theorem 3.6. If $(d_{2},q_{2})\in
G_{23}^{*}$, then $q_{2}<\underline{q}$ and $(d_{1},q_{1})\in\hat{G}_{13}$,
where

$\hat{G}_{13}:=\\{(d,q):\;d\geq d_{2},q_{2}\leq
q<\frac{q_{2}}{d_{2}}d,(d,q)\neq(d_{2},q_{2})\\}.$

Since the nonlinear terms of model (1.4) are symmetric, it follows from
Theorem 3.6 (ii) that $(\bm{0},\bm{v}^{*})$ is globally asymptotically stable.

## Appendix

In the appendix, we study the relations of $\overline{q}$, $\underline{q}$ and
$q_{0}$. For convenience, we recall the definition of $\overline{q}$,
$\underline{q}$ and $q_{0}$:

$\displaystyle\overline{q}=\max\left\\{\displaystyle\frac{r_{1}}{k_{1}}(k_{1}-k_{2}),\displaystyle\frac{r_{3}}{k_{3}}(k_{2}-k_{3})\right\\},$
(6.9a)
$\displaystyle\underline{q}=\min\left\\{\displaystyle\frac{r_{1}}{k_{1}}(k_{1}-k_{2}),\displaystyle\frac{r_{3}}{k_{3}}(k_{2}-k_{3})\right\\},$
(6.9b) $\displaystyle
q_{0}=\max\left\\{r_{1}\left(1-\frac{u_{1}^{*}}{k_{1}}\right),r_{2}\left(1-\frac{u_{2}^{*}}{k_{2}}\right)\right\\}.$
(6.9c)

######  Lemma 6.4.

Suppose that $(\bf{H})$ holds, $\bm{r}\gg\bm{0}$, and $d_{1},q_{1}>0$. Then
the following statements hold:

  1. (i)

If $q_{1}<\underline{q}$, then $q_{0}>q_{1}$;

  2. (ii)

If $q_{1}>\overline{q}$, then $q_{0}<q_{1}$;

  3. (iii)

If $q_{1}>\underline{q}$, then $q_{0}>\underline{q}$;

  4. (iv)

If $q_{1}<\overline{q}$, then $q_{0}<\overline{q}$.

###### Proof.

By (5.10)-(5.11) and Lemma 5.1 (i), we have

$\displaystyle\tilde{f}_{1}=d_{1}u_{2}^{*}-(d_{1}+q_{1})u_{1}^{*}=-r_{1}u_{1}^{*}\left(1-\frac{u_{1}^{*}}{k_{1}}\right)<0,$
(6.10a)
$\displaystyle\tilde{f}_{2}=d_{1}u_{3}^{*}-(d_{2}+q_{2})u_{2}^{*}=r_{3}\left(1-\frac{u_{3}^{*}}{k_{3}}\right)<0,$
(6.10b)
$\displaystyle\tilde{f}_{2}-\tilde{f}_{1}=-r_{2}u_{2}^{*}\left(1-\frac{u_{2}^{*}}{k_{2}}\right),$
(6.10c)

which will be used in the proof below.

(i) By Lemma 5.1 (iv), we have $u^{*}_{1}>u^{*}_{2}>u^{*}_{3}$. This, together
with (6.9c) and (6.10a), implies that

$q_{0}\geq
r_{1}\left(1-\frac{u_{1}^{*}}{k_{1}}\right)=\frac{(d_{1}+q_{1})u^{*}_{1}-d_{1}u^{*}_{2}}{u^{*}_{1}}=d_{1}\left(1-\frac{u^{*}_{2}}{u^{*}_{1}}\right)+q_{1}>q_{1}.$
(6.11)

(ii) By Lemma 5.1 (iii), we have $u^{*}_{1}<u^{*}_{2}<u^{*}_{3}$. Then by
(6.10a) again, we obtain

$r_{1}\left(1-\frac{u_{1}^{*}}{k_{1}}\right)=d_{1}\left(1-\frac{u^{*}_{2}}{u^{*}_{1}}\right)+q_{1}<q_{1}.$
(6.12)

By (6.10c), we obtain that

$\begin{split}r_{2}\left(1-\frac{u_{2}^{*}}{k_{2}}\right)=&\displaystyle\frac{\left((d_{1}+q_{1})u^{*}_{2}-d_{1}u_{3}^{*}\right)+\tilde{f}_{1}}{u^{*}_{2}}\\\
=&d_{1}\left(1-\frac{u^{*}_{3}}{u^{*}_{2}}\right)+\frac{\tilde{f}_{1}}{u^{*}_{2}}+q_{1}<q_{1},\end{split}$
(6.13)

where we have used (6.10c) and $u_{2}^{*}<u_{3}^{*}$ in the last step. It
follows from (6.9c), (6.12) and (6.13) that $q_{0}<q_{1}$.

(iii) We divide the proof into three cases:

$({\rm A1})\;\;u_{1}^{*}<k_{2},\;\;({\rm A2})\;\;u^{*}_{1}\geq
u^{*}_{2},\;\;({\rm A3})\;\;k_{2}\leq u_{1}^{*}<u_{2}^{*}.$

For case (A1), we see from (6.9b) and (6.9c) that

$q_{0}\geq
r_{1}\left(1-\frac{u_{1}^{*}}{k_{1}}\right)>\displaystyle\frac{r_{1}}{k_{1}}(k_{1}-k_{2})\geq\underline{q}.$

For case (A2), we see from (6.9c) and (6.10a) that

$q_{0}\geq
r_{1}\left(1-\frac{u_{1}^{*}}{k_{1}}\right)=d_{1}\left(1-\frac{u^{*}_{2}}{u^{*}_{1}}\right)+q_{1}\geq
q_{1}>\underline{q}.$

Now we consider (A3). Suppose to the contrary that $q_{0}\leq\underline{q}$.
This, combined with (6.9b)-(6.9c), yields

$r_{1}\left(1-\frac{u_{1}^{*}}{k_{1}}\right)\leq
q_{0}\leq\underline{q}\leq\frac{r_{3}}{k_{3}}\left(k_{2}-k_{3}\right).$ (6.14)

Noticing that $u_{2}^{*}>k_{2}$, we see from (6.10c) that

$\tilde{f}_{2}-\tilde{f}_{1}=d_{1}(u^{*}_{3}-u^{*}_{2})-(d_{1}+q_{1})(u^{*}_{2}-u^{*}_{1})=-r_{2}u^{*}_{2}\left(1-\displaystyle\frac{u^{*}_{2}}{k_{2}}\right)>0.$
(6.15)

Since $u^{*}_{1}<u^{*}_{2}$, we see from (6.15) that $u^{*}_{2}<u^{*}_{3}$.
Then we have

$0>\tilde{f}_{2}>\tilde{f}_{1}\;\;\text{and}\;\;k_{2}\leq
u^{*}_{1}<u^{*}_{2}<u^{*}_{3},$ (6.16)

which yields

$-\displaystyle\frac{\tilde{f}_{2}}{u^{*}_{3}}<-\displaystyle\frac{\tilde{f}_{1}}{u^{*}_{1}}.$
(6.17)

This, together with (6.16), (6.10a) and (6.10b), implies that

$r_{1}\left(1-\frac{u^{*}_{1}}{k_{1}}\right)=-\displaystyle\frac{\tilde{f}_{1}}{u^{*}_{1}}>-\displaystyle\frac{\tilde{f}_{2}}{u^{*}_{3}}=\displaystyle\frac{r_{3}}{k_{3}}\left(u^{*}_{3}-k_{3}\right)>\displaystyle\frac{r_{3}}{k_{3}}\left(k_{2}-k_{3}\right),$

which contradicts (6.14). Therefore, $q_{0}>\underline{q}$ for case (A3).

(iv) We first show that

$r_{1}\left(1-\frac{u_{1}^{*}}{k_{1}}\right)<\overline{q},$ (6.18)

and the proof is divided into three cases:

$({\rm B1})\;\;u^{*}_{1}>k_{2},\;\;({\rm B2})\;\;u_{1}^{*}\leq
u_{2}^{*},\;\;({\rm B3})\;\;k_{2}\geq u^{*}_{1}>u^{*}_{2}.$

For case (B1), we have

$r_{1}\left(1-\frac{u_{1}^{*}}{k_{1}}\right)<\frac{r_{1}}{k_{1}}(k_{1}-k_{2})\leq\overline{q}.$

For case (B2), we see from (6.10a) that

$r_{1}\left(1-\frac{u_{1}^{*}}{k_{1}}\right)=d_{1}\left(1-\frac{u^{*}_{2}}{u^{*}_{1}}\right)+q_{1}\leq
q_{1}<\overline{q}.$ (6.19)

For case (B3), using similar arguments as the above case (A3), we have

$0>\tilde{f}_{1}>\tilde{f}_{2}\;\;\text{and}\;\;k_{2}\geq
u^{*}_{1}>u^{*}_{2}>u^{*}_{3}.$

This, combined with (6.10a) and (6.10b), implies that

$r_{1}\left(1-\frac{u_{1}^{*}}{k_{1}}\right)=-\frac{\tilde{f}_{1}}{u^{*}_{1}}<-\frac{\tilde{f}_{2}}{u^{*}_{1}}<-\frac{\tilde{f}_{2}}{u^{*}_{3}}=r_{3}\left(\frac{u_{3}^{*}}{k_{3}}-1\right)<\frac{r_{3}}{k_{3}}(k_{2}-k_{3})\leq\overline{q}.$

Then we show that

$r_{2}\left(1-\frac{u_{2}^{*}}{k_{2}}\right)<\overline{q},$ (6.20)

and the proof is also divided into three cases:

$({\rm C1})\;\;u^{*}_{2}\leq u_{3}^{*},\;\;({\rm
C2})\;\;u_{2}^{*}>u_{3}^{*}\geq k_{2},\;\;({\rm
C3})\;\;u_{2}^{*}>u_{3}^{*}\;\;\text{and}\;\;k_{2}>u_{3}^{*}.$

For case (C1), we see from (6.13) that

$r_{2}\left(1-\frac{u_{2}^{*}}{k_{2}}\right)<q_{1}<\overline{q}.$

For case (C2), we have

$r_{2}\left(1-\frac{u_{2}^{*}}{k_{2}}\right)<0<\overline{q}.$

For case (C3), we see from (6.10) that

$r_{2}\left(1-\frac{u_{2}^{*}}{k_{2}}\right)=\frac{\tilde{f}_{1}-\tilde{f}_{2}}{u^{*}_{2}}<-\frac{\tilde{f}_{2}}{u^{*}_{2}}<-\frac{\tilde{f}_{2}}{u^{*}_{3}}=\frac{r_{3}}{k_{3}}\left(u^{*}_{3}-k_{3}\right)<\frac{r_{3}}{k_{3}}\left(k_{2}-k_{3}\right)\leq\overline{q}.$

By (6.18) and (6.20), we see that (iv) holds. ∎

######  Remark 6.5.

By $\underline{q}\leq\overline{q}$ and Lemma 6.4, we see that if
$q_{1}<\underline{q}$, then $q_{1}<q_{0}<\overline{q}$; if
$q_{1}>\overline{q}$, then $\underline{q}<q_{0}<q_{1}$; and if
$\underline{q}<q_{1}<\overline{q}$, then $\underline{q}<q_{0}<\overline{q}$.

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