Anushaya Mohapatra]

Anushaya Mohapatra]

Anushaya Mohapatra]

# The Evolutionary stability of partial migration with Allee effects

Yogesh Trivedi  Department of Mathematics
BITS Pilani, K.K Birla Goa Campus, 403726, Goa, India [ ,  Ram Singh
Department of Mathematics
BITS Pilani, K.K Birla Goa Campus, 403726, Goa, India [ and  Anushaya
Mohapatra  Department of Mathematics
BITS Pilani, K.K Birla Goa Campus, 403726, Goa, India [

###### Abstract.

An Allee effect occurs when the per-capita growth rate increases at low
densities. Here, we investigate the evolutionary stability of a partial
migration population with migrant population experiencing Allee effects.
Partial migration is a unique form of phenotypic diversity wherein migrant and
non-migrant individuals coexist together. It is shown that when Allee effect
is incorporated, the population undergoes a bifurcation as the fraction of
migrating population increases from zero to unity. Using an evolutionary game
theoretic approach, we prove the existence of a unique evolutionary stable
strategy (ESS). It is also shown that the ESS is the only ideal free
distribution (IFD) that arises in the context of partially migrating
population.

###### Key words and phrases:

Partial Migration, Allee Effects, Basic Reproduction Number, Ideal Free
Distribution, Evolutionary Game Theory, Evolutionary Stable Strategy

###### 2000 Mathematics Subject Classification:

37

##  1\. Introduction

Allee effect is a phenomenon, in which individual fitness increases with
increasing density at low densities. Many research has shown the evidence of
Allee effects and the importance of this biological phenomenon has been widely
recognized in population dynamics, conservation programs, management of
endangered species and ecosystem dynamics [2, 7, 10, 11, 12, 16]. Here we
study the evolution of partial migration phenomenon when only migrants
experience Allee effects. Migration is a diverse phenomenon, and can be
categorized into a multitude of forms. The most common type of migration is
known as partial migration in which some individual migrate between habitats
and others remain in a single habitat during their entire life. Originally,
these studies were motivated by birds, by now, partial migration has been
found across many taxa, including fish, invertebrates and mammals [6, 19, 20,
23, 24]. This type of within-population diversity is thought to play an
important role in population stability and resilience [31], so understanding
how it is maintained by natural selection is critical for predicting how
species may respond to future conditions.

There has been lot of interest in understanding such a complex system in which
population consists of a mix of migratory and non-migratory individuals.
Several mechanisms are studied in association with partial migration which
include genetic control, density-dependence, and exogenous stochastic effects
in environmental variables [6, 17, 23, 29, 30]. In general, partial migration
population models with negative density dependence effects are studied and
neglecting the populations that experience Allee effects [17, 21, 26, 29]. One
of the most common examples of Allee effects occur when a species is subject
to predation with a saturating functional response, meaning that increased
population levels decrease the risk of predation. Motivated by the partially
migrating fish population steelhead rainbow trout system, in which steelhead
experiences predation during its ocean phase, but trout does not, and only
experiences the usual negative density dependence, we study a partial
migration population model where the critical component is incorporation of an
Allee effect. The primary goal of this work is to investigate what will happen
when the biological mechanism Allee effects is included. Is partial migration
preserved? Or can it be lost? How robust is it when the underlying population
model is modified to incorporate the neglected biological feature?

We investigate the evolutionary stability of the partial migration population
with strong Allee effect (migrants) using evolutionary game theoretic or
Darwin’s dynamics approach. This is an alternative approach to Adaptive
dynamics and is based on Evolutionary Game Theory as given in [19, 20, 27,
32]. We also explore the connections between Ideal Free Distributions (IFD)
arising from strategies that lead to partial migration behavior and
evolutionary stable startegies. It turns out that the strategy which is the
fraction of population became migrant act as a bifurcation parameter for the
model. More ever it is shown that under rather general conditions the only
possible strategies corresponding to an IFD, are evolutionary stable, as well
as convergent stable.

The rest of this paper is structured as follows. In Section 2 we formulate the
population model and state the stability results. In Section 3 we derive an
explicit formula for the ESS using evolutionary game approach. In Section 4 we
discuss the results that connect IFDs and ESSs, and we conclude in Section 5
with possible future work and a brief non-mathematical discussion of the
results obtained in this research.

##  2\. The population model with strong Allee effect

Consider the following stage structured, density-dependent and matrix
population model.

(1) $\begin{pmatrix}x_{1}(t+1)\\\ x_{M}(t+1)\\\
x_{N}(t+1)\end{pmatrix}=\begin{pmatrix}0&f_{M}(z_{M}(t))&f_{N}(z_{N}(t))\\\
\phi s_{M}&0&0\\\ (1-\phi)s_{N}&0&0\end{pmatrix}\begin{pmatrix}x_{1}(t)\\\
x_{M}(t)\\\ x_{N}(t)\end{pmatrix},$

where $x_{1}(t)$, $x_{M}(t)$ and $x_{N}(t)$ are non-negative real numbers,
respectively representing the abundances of eggs, migrant adults and non-
migrant adults at time $t$, where $t$ is a non-negative integer. A fraction
$\phi\in[0,1]$ of eggs at time $t$ will become migrant adults, provided they
survive a season, which is captured by the survival probability
$s_{M}\in(0,1]$ in the model. Similarly, a fraction $1-\phi$ of eggs will
become non-migrant adults, after surviving a season, with survival probability
$s_{N}\in(0,1]$. The functions $f_{M}(z)$ and $f_{N}(z)$ are per capita
fertilities of migrants and non-migrants respectively. The parameter $\phi$
represents an allocation strategy whereby each morph (migrant or non-migrant)
produces offspring that can become either type of morphs. $z_{i}$, with
$i=M,N$ represents the total number of competing individuals experienced by
phenotype $i$ during reproduction, and it is given by

(2) $z_{M}(t)=x_{M}(t)+px_{N}(t)$

and

(3) $z_{N}(t)=x_{N}(t)+qx_{M}(t)$

with $0<p,q<1$. Here, $p$ is a parameter representing the fraction of the non-
migrant population competing with each migrant adult. Similarly $q$ represents
the fraction of the migrant population that competes with each non-migrant
adult.

We now describe the assumptions behind our general results:

  1. (H1)

(Migrant Allee effect) $f_{M}:[0,\infty)\to[0,\infty)$ is a smooth and
unimodal function. Namely there is an unique positive population density
$C_{0}$ such that $f^{{}^{\prime}}_{M}(C_{0})=0$ and the maximum value of the
function is $f_{M}(C_{0})$. $f_{M}$ increases at low densities i.e
$f_{M}^{{}^{\prime}}(z)>0$ for z sufficiently small. There is a positive
equilibrium density A such that $f_{M}(z)<1$ for all $z<A$ and $f_{M}(z)>1$
for some $z>A$. More ever we choose $s_{M}f_{M}(C_{0})>1$ and
$g_{M}(z)=zf_{M}(z)$ has positive derivative.

######  Remark 2.1.

In the case of $s_{M}f_{M}(C_{0})<1$, the migrant only population model has
only zero equilibrium point that may lead to extinction. So the assumption
$s_{M}f_{M}(C_{0})>1$ guarantee the existence of non-zero or positive fixed
points for migrant only population. The solution set of the equation
$s_{M}f_{M}(z)=1$ is non-empty, more ever it has precisely two element say
$\hat{x}_{M}$ and $\tilde{x}_{M}$ with $0<\tilde{x}_{M}<C_{0}<\hat{x}_{M}$,
where $\hat{x}_{M}$ is referred as the carrying capacity of the migrant
population.

######  Example 2.2.

The most common Allee effect occurs in species subject to predation by a
generalist predator with a saturating functional response. Migrant population
can be modeled by the following equation

(4) $f_{M}(z)=\frac{az}{(b+z)^{2}}$

If we choose $a=5$ and $b=1$ then the graph of the function $f_{M}(z)$ is as
shown in the following figure.

Figure 1. The function $f_{M}$

  2. (H2)

(Decreasing fertility ) $f_{N}:[0,\infty)\to(0,\infty)$ is a smooth function,
with negative derivatives and $g_{N}(z):=f_{N}(z)z$ has positive derivative.
More ever $s_{N}f_{N}(0)>1$. Biologically, reproduction decrease with
increasing population density.

######  Remark 2.3.

The condition $s_{N}f_{N}(0)>1$ imply that when only non-migrant morph is
present, the population persists. The equation $s_{N}f_{N}(z)=1$ has a unique
solution $\tilde{x}_{N}>0$ and referred as the carrying capacity of the non-
migrant population.

###  2.1. Stability Analysis

Model $(\ref{sys-1})$ can be re-written more compactly in vector form as

(5) $X(t+1)=A_{1}(X(t),\phi)X(t),$

where

$X=\begin{pmatrix}x_{1}\\\ x_{M}\\\
x_{N}\end{pmatrix},A_{1}(X,\phi)=\begin{pmatrix}0&f_{M}(z_{M})&f_{N}(z_{N})\\\
\phi s_{M}&0&0\\\ (1-\phi)s_{N}&0&0\end{pmatrix},$

$z_{M}(t)=x_{M}(t)+px_{N}(t)$ and $z_{N}(t)=x_{N}(t)+qx_{M}(t).$ By splitting
$A_{1}(X,\phi)$ as:

$A_{1}(X,\phi)=F+T,\textrm{ where
}F=\begin{pmatrix}0&f_{M}(z_{M})&f_{N}(z_{N})\\\ 0&0&0\\\
0&0&0\end{pmatrix},\textrm{ and }T=\begin{pmatrix}0&0&0\\\ \phi s_{M}&0&0\\\
(1-\phi)s_{N}&0&0\end{pmatrix},$

allen2008basic,caswell2000matrix,cushing1998introduction,li2002applications we
can associate the basic reproduction number to the non-negative matrix
$A_{1}(X,\phi)$ in the usual way [1, 5, 9, 22]:

(6) $R_{0}(X,\phi):=\rho(F(I-T)^{-1})=\phi
R_{0}^{M}(z_{M})+(1-\phi)R_{0}^{N}(z_{N})\textrm{ for every
}(X,\phi)\in\mathbb{R}^{3}_{+}\times[0,1].$

Here, $\rho(F(I-T)^{-1})$ denotes the spectral radius of $F(I-T)^{-1}$ and
$R_{0}^{I}(z_{I})=s_{I}f_{I}(z_{I})$ for $I=M,N.$

######  Lemma 2.4.

Suppose $\phi_{1}=\frac{1-s_{N}f_{N}(0)}{s_{M}f_{M}(0)-s_{N}f_{N}(0)}$, then
for $\phi\in[0,\phi_{1}),R_{0}(0,\phi)>1$, and for
$\phi\in(\phi_{1},1],R_{0}(0,\phi)<1$.

###### Proof.

We have $R_{0}(0,\phi)=\phi s_{M}f_{M}(0)+(1-\phi)s_{N}f_{N}(0)$. The
derivative of $R_{0}(0,\phi)$ w.r.t $\phi$ is $s_{M}f_{M}(0)-s_{N}f_{N}(0)$
which is negative as $s_{M}f_{M}(0)<1<s_{N}f_{N}(0)$ by hypothesis (H1) and
(H2). So $R_{0}(0,\phi)$ is a decreasing function of $\phi$. More ever we have
$R_{0}(0,\phi)=1$ iff $\phi=\phi_{1}$, where
$\phi_{1}=\frac{1-s_{N}f_{N}(0)}{s_{M}f_{M}(0)-s_{N}f_{N}(0)}.$ Hence for
$\phi\in[0,\phi_{1}),R_{0}(0,\phi)>1$ and for
$\phi\in(\phi_{1},1],R_{0}(0,\phi)<1.$ ∎

######  Lemma 2.5.

Suppose the functions $f_{M}$ and $f_{N}$ satisfies the hypothesis
$({\bf\ref{Allee-type}})-({\bf H2})$ and suppose that
$s_{N}f_{N}(\frac{C_{0}}{p})>1,qs_{M}<s_{N}$ and $ps_{N}<s_{M}$ . Define
$Q_{\phi}(x):=\phi s_{M}f_{M}(d_{1}x)+(1-\phi)s_{N}f_{N}(d_{2}x)$ where
$d_{1}(\phi)=\phi s_{M}+p(1-\phi)s_{N}$ and $d_{2}(\phi)=q\phi
s_{M}+(1-\phi)s_{N}$. Exactly one solution ${\bar{x}_{1}}(\phi)$ of the
equation $Q_{\phi}(x)=1$ will lie in the interval
$(\frac{C_{0}}{d_{1}(\phi)},\infty)$, where $f_{M}^{{}^{\prime}}(C_{0})=0$.

###### Proof.

Denote $F_{\phi}^{M}(x)=1-\phi s_{M}f_{M}(d_{1}x)$ and
$F_{\phi}^{N}(x)=(1-\phi)s_{N}f_{N}(d_{2}x)$, then we have

$Q_{\phi}(x)-1=F_{\phi}^{N}(x)-F_{\phi}^{M}(x)$

Notice that $\frac{d}{dx}(F^{M}_{\phi})=0$ at $\frac{C_{0}}{d_{1}}$ and
$F_{\phi}^{M}$ is a unimodal function for each $\phi\in(0,1)$ as shown in the
figure (2). Also $F_{\phi}^{N}$ is a strictly decreasing function in x. So if
$F_{\phi}^{M}(\frac{C_{0}}{d_{1}})<F_{\phi}^{N}({\frac{C_{0}}{d_{1}}})$, then
the graphs of $F_{\phi}^{M}$ and $F_{\phi}^{N}$ will intersect at least at one
point in the first quadrant. It is easy to check that the assumptions
$s_{N}f_{N}(\frac{C_{0}}{p})>1,qs_{M}<s_{N}$, $ps_{N}<s_{M}$ and
$s_{M}f_{M}(C_{0})>1$ together implies
$F_{\phi}^{M}(\frac{C_{0}}{d_{1}})<F_{\phi}^{N}({\frac{C_{0}}{d_{1}}})$ . So
the equation $Q_{\phi}(x)=1$ has at least one solution in positive real line.
Since $F^{N}_{\phi}$ is strictly decreasing and $F^{M}_{\phi}$ is strictly
increasing on $[\frac{C_{0}}{d_{1}},\infty)$, the equation $Q_{\phi}(x)=1$ has
exactly one solution $\bar{x}_{1}(\phi)$ in $(\frac{C_{0}}{d_{1}},\infty)$
(see figure 2.) ∎

######  Remark 2.6.

Lemma (2.5) says that the number of positive solution of the equation
$Q_{\phi}(x)=1$ in $(\frac{C_{0}}{d_{1}(\phi)},\infty)$ is one, however there
may be one or more then one positive solution of the equation $Q_{\phi}(x)=1$
in the interval $[0,\frac{C_{0}}{d_{1}})$. For simplification purpose we only
focus on a class functions $f_{N}$, $f_{M}$ and other suitable parameters such
that the equation $Q_{\phi}(x)=1$ has at most one solution in
$[0,\frac{C_{0}}{d_{1}})$ for all $\phi\in(0,1)$ as shown in the figure (2)
and also the main results in this work remain same even if the equation
$Q_{\phi}(x)=1$ has one or more solutions in $[0,\frac{C_{0}}{d_{1}})$.

(a) $\phi=\phi_{1}$

(b) $\phi\in(0,\phi_{1})$

(c) $\phi\in(\phi_{1},1)$

Figure 2.

######  Remark 2.7.

The assumption $s_{N}f_{N}(\frac{C_{0}}{p})>1$ in lemma (2.5) is equivalent to
$C_{0}<p\tilde{x}_{N}$, provided $qs_{M}<s_{N}$ and $ps_{N}<s_{M}$, where
$f_{M}^{{}^{\prime}}(C_{0})=0$.

So we make another hypothesis as follows.

  3. (H3)

For each $\phi\in(0,1)$ the equation $Q_{\phi}(x)=1$ has at most one positive
solution in the interval $(0,\frac{C_{0}}{d_{1}})$ as shown in the figure (2),
where $Q_{\phi}(x)$ is the function as defined in lemma (2.5). And more ever
as assumed in the lemma (2.5) choose $C_{0}<p\tilde{x}_{N},qs_{M}<s_{N}$ and
$ps_{N}<s_{M}$.

######  Example 2.8.

Let $f_{M}(z)=\frac{7}{(1+z)^{2}}$ , $f_{N}(z)=\frac{3}{1+z}$ and
$s_{M}=0.70,s_{N}=0.85,q=0.80,p=0.80$. In this case we get $\phi_{1}=0.61$, so
when $\phi\in[0,0.61)$ i.e. $R_{0}(0,\phi)>1$, the functions $F_{\phi}^{M}$
and $F_{\phi}^{N}$intersect at exactly one point. And if $\phi\in(0.61,1]$
i.e. $R_{0}(0,\phi)<1$, the functions $F_{\phi}^{M}$ and $F_{\phi}^{N}$
intersect at two points. Hence the equation $Q_{\phi}(x)=1$ has one positive
solution for $R_{0}(0,\phi)>1$, and two positive solutions for
$R_{0}(0,\phi)<1$. The graphs corresponding to some $\phi$ values are given
below. The functions $f_{N}$ , $f_{M}$ and other parameters satisfies all the
hypothesis (H1) \- (H3).

(a) $\phi=0.4$ i.e. $R_{0}(0,\phi)>1$

(b) $\phi=0.75$ i.e. $R_{0}(0,\phi)<1$

Figure 3.

######  Example 2.9.

Let $f_{M}(z)=\frac{8}{(1+z)^{2}}$ , $f_{N}(z)=\frac{9}{1+1.2z}$ and
$s_{M}=0.60,s_{N}=0.28,q=0.45,p=0.90$. In this case we get $\phi_{1}=0.60$, so
when $\phi\in[0,0.60)$ i.e. $R_{0}(0,\phi)>1$, the functions $F_{\phi}^{M}$
and $F_{\phi}^{N}$ intersect at exactly one point. And if $\phi\in(0.60,1]$
i.e. $R_{0}(0,\phi)<1$, the functions $F_{\phi}^{M}$ and $F_{\phi}^{N}$
intersect at two points. Hence the equation $Q_{\phi}(x)=1$ has one positive
solution for $R_{0}(0,\phi)>1$, and two positive solutions for
$R_{0}(0,\phi)<1$. The graphs corresponding to some $\phi$ values are given
below. The functions $f_{N}$ , $f_{M}$ and other parameters satisfies all the
hypothesis (H1) \- (H3)

(a) $\phi=0.50$ i.e. $R_{0}(0,\phi)>1$

(b) $\phi=0.85$ i.e. $R_{0}(0,\phi)<1$

Figure 4.

###  2.2. The population dynamics

One can observe that if there exist a positive fixed point
$x^{*}=(x_{1}^{*},x_{M}^{*},x_{N}^{*})$ of the system (5), then it should
satisfy the equation $R_{0}(\phi,x^{*})=1$. If we choose $d_{1}=\phi
s_{M}+p(1-\phi)s_{N}$ and $d_{2}=q\phi s_{M}+(1-\phi)s_{N}$, then the first
coordinate $x_{1}^{*}$ satisfy the following

(7) $Q_{\phi}(x)=\phi s_{M}f_{M}(d_{1}x)+(1-\phi)s_{N}f_{N}(d_{2}x)=1$

Conversely suppose any positive $x_{1}^{*}$ satisfy the equation (7). If you
choose $x^{*}=(x_{1}^{*},x_{M}^{*},x_{N}^{*})$ with $x_{M}^{*}=\phi
s_{M}x_{1}^{*}$ and $x_{N}^{*}=(1-\phi)s_{N}x_{1}^{*}$, then
$R_{0}(x^{*},\phi)=1$ and $x^{*}$ is a positive fixed point of the model (5).
Since $x^{*}$ is uniquely defined by $x_{1}^{*}$, the number of positive
solution to the equation (7) is same as the number of positive fixed point of
the system (5).

We have the following result.

######  Theorem 2.10.

Assume that the hypothesis (H1) \- (H3) holds for the system (1) then
following holds with
$\phi_{1}=\frac{1-s_{N}f_{N}(0)}{s_{M}f_{M}(0)-s_{N}f_{N}(0)}$:

  1. (1)

For $\phi\in[0,\phi_{1})$, the zero fixed point is unstable and for
$\phi\in(\phi_{1},1]$, it is locally stable.

  2. (2)

If $\phi\in(0,1)$, the system (1) has exactly one locally stable positive
fixed point $x^{*}(\phi)=(x_{1}^{*},x_{M}^{*},x_{N}^{*})$ with
$x_{1}^{*}>\frac{C_{0}}{d_{1}}$, where $d_{1}=\phi s_{M}+p(1-\phi)s_{N}$ and
$f_{M}^{{}^{\prime}}(C_{0})=0.$

  3. (3)

If $\phi=0$, then system $(\ref{sys-1})$ has a unique, non-zero locally stable
fixed point $({\tilde{x}}_{1},0,{\tilde{x}}_{N})$, where ${\tilde{x}}_{N}>0$
is the unique positive solution to the equation $s_{N}f_{N}(z)=1$, and If
$\phi=1$, then system $(\ref{sys-1})$ has exactly one non-zero locally stable
fixed point $({\hat{x}}_{1},{\hat{x}}_{M},0)$, where ${\hat{x}}_{M}$ is a
positive solution to the equation $s_{M}f_{M}(z)=1$, with
$f_{M}^{{}^{\prime}}({\hat{x}}_{M})<0$ and
${\hat{x}}_{1}={\hat{x}}_{M}/s_{M}$, ${\tilde{x}}_{1}={\tilde{x}}_{N}/s_{N}$.

###### Proof.

For $\phi\in[0,\phi_{1})$, $R_{0}(0,\phi)>1$ and for
$\phi\in(\phi_{1},1],R_{0}(0,\phi)<1$ by Lemma (2.4). So for
$\phi\in(\phi_{1},1]$ the origin is a locally stable fixed point and for
$\phi\in[0,\phi_{1})$, it is unstable.

Case I : $\phi\in(0,1).$
Suppose $R_{0}(0,\phi)\geq 1$, then by lemma (2.5) and hypothesis
$(\textbf{H}3)$, for each $\phi$ the system (1) has one positive fixed point
$x^{*}(\phi)=(x_{1}^{*},x_{M}^{*},x_{N}^{*})$ with
$x_{1}^{*}>\frac{C_{0}}{d_{1}}$. The Jacobian matrix $J$ defined at the fixed
point $(x_{1}^{*},x_{M}^{*},x_{N}^{*})$ is

(8) $J=\begin{pmatrix}0&F_{1}&F_{2}\\\ \phi s_{M}&0&0\\\
(1-\phi)s_{N}&0&0\end{pmatrix}$

where
$F_{1}=x_{M}^{*}{(t)}f^{\prime}_{M}(z_{M}^{*}{(t)})+f_{M}(z_{M}^{*}{(t)})+qx_{N}^{*}{(t)}f^{\prime}_{N}(z_{N}^{*}{(t)})$
and
$F_{2}=px_{M}^{*}{(t)}f^{\prime}_{M}(z_{M}^{*}{(t)})+x_{N}^{*}{(t)}f^{\prime}_{N}(z_{N}^{*}{(t)})+f_{N}(z_{N}^{*}{(t)})$.
The eigen values of the matrix J are found to be $0$, and
$\pm\sqrt{1+x_{1}^{*}.Q_{\phi}^{\prime}(x_{1}^{*})}$. Where

$Q_{\phi}^{\prime}(x_{1}^{*})=d_{1}\phi
s_{M}f^{\prime}_{M}(d_{1}x_{1}^{*}{(t)})+d_{2}(1-\phi)s_{N}f^{\prime}_{N}(d_{2}x_{1}^{*}{(t)})$

with $d_{1}=\phi s_{M}+p(1-\phi)s_{N}$ and $d_{2}=q\phi s_{M}+(1-\phi)s_{N}$.
Since the derivative of the functions $xf_{M}(d_{1}x)$ and $xf_{N}(d_{2}x)$
are positive, we have $\lvert
d_{1}x_{1}^{*}f^{\prime}_{M}(d_{1}x_{1}^{*})\rvert<\lvert
f_{M}(d_{1}x_{1}^{*})\rvert$ and $\lvert
d_{2}x_{1}^{*}f^{\prime}_{N}(d_{2}x_{1}^{*})\rvert<\lvert
f_{N}(d_{2}x_{1}^{*})\rvert$. So $\lvert
x_{1}^{*}.Q_{\phi}^{\prime}(x_{1}^{*})\rvert<\lvert d_{1}\phi
s_{M}x_{1}^{*}f^{\prime}_{M}(d_{1}x_{1}^{*})\rvert+\lvert
d_{2}(1-\phi)s_{N}x_{1}^{*}f^{\prime}_{N}(d_{2}x_{1}^{*})\rvert<1$. So the
eigen value $\lambda=|\sqrt{1+x_{1}^{*}.Q_{\phi}^{\prime}(x_{1}^{*})}|$ of the
matrix J is dominant, positive and less than one as
$x_{1}^{*}Q_{\phi}^{\prime}(x_{1}^{*})<0$. Hence the fixed point
$(x_{1}^{*},x_{M}^{*},x_{N}^{*})$ is locally asymptotically stable. Now
suppose $R_{0}(0,\phi)<1$, by lemma (2.5) and hypothesis $(\textbf{H}3)$, for
each $\phi$ the system (1) has two positive fixed points say $x^{**}(\phi)$
and $x^{*}(\phi)$. Suppose $x_{1}^{**}$ and $x_{1}^{*}$ are the corresponding
first coordinates with $x_{1}^{**}<\frac{C_{0}}{d_{1}}<x_{1}^{*}$. Similar to
the previous case (for $R_{0}(0,\phi)\geq 1$) the eigen values of the Jacobian
of the system (1) at $x^{**}$ are $0,$ and
$\pm\sqrt{1+x_{1}^{**}.Q_{\phi}^{\prime}(x_{1}^{**})}$. In this case
$x_{1}^{**}.Q_{\phi}^{\prime}(x_{1}^{**})$ is positive as $Q_{\phi}(x)-1$ is
an increasing function at $x_{1}^{**}$, so the eigen value
$\lambda=|\sqrt{1+x_{1}^{**}.Q_{\phi}^{\prime}(x_{1}^{**})}|$ is dominant,
positive and bigger than one which make the fixed point $x^{**}(\phi)$ locally
asymptotically unstable . On the other hand $x^{*}$ is locally stable using
the same argument as in the case of $R_{0}(0,\phi)\geq 1$ . So for each
$\phi\in(0,1)$, the system (1) has exactly one locally stable fixed point
$(x_{1}^{*}(\phi),x_{M}^{*}(\phi),x_{N}^{*}(\phi))$.

Case II : Suppose $\phi=0$, then notice that every orbit of $(\ref{sys-1})$
enters the invariant part of the the boundary of $\mathbb{R}^{3}_{+}$ where
$x_{M}=0$ in $1$ step. The restriction of the dynamics to this part of the
boundary is given by a planar system:

(9) $\begin{pmatrix}x_{1}(t+1)\\\
x_{N}(t+1)\end{pmatrix}=\begin{pmatrix}0&f_{N}(x_{N}(t))\\\
s_{N}&0\end{pmatrix}\begin{pmatrix}x_{1}(t)\\\ x_{N}(t)\end{pmatrix}$

Suppose that $(x_{1}^{*},x_{N}^{*})$ is a positive fixed point of the system
(9) then it turns out that $s_{N}f_{N}(x_{N}^{*}{(t)})=1$ which has a unique
positive solution say ${\tilde{x}}_{N}$. Choose
${\tilde{x}}_{1}=\frac{{\tilde{x}}_{N}}{s_{N}}$, clearly
$({\tilde{x}}_{1},0,{\tilde{x}}_{N})$ the unique non-zero fixed point of the
system (1) when $\phi=0$. The Jacobian matrix when we linearize the system (9)
near the fixed point $(\frac{{\tilde{x}}_{N}}{s_{N}},{\tilde{x}}_{N})$ is
given by

(10)
$J_{1}=\begin{pmatrix}0&{\tilde{x}}_{N}f_{N}{{}^{\prime}}({\tilde{x}}_{N})+f_{N}({\tilde{x}}_{N})\\\
s_{N}&0\end{pmatrix}$

The eigen values of the matrix $J_{1}$ are given by
$\lambda=\pm\sqrt{1+s_{N}{\tilde{x}}_{N}f^{\prime}_{N}({\tilde{x}}_{N})}$.
Since the function $zf_{N}(z)$ is increasing and $f_{N}(z)$ is decreasing,
$\mid\lambda\mid<1$, so the fixed point $({\tilde{x}}_{1},0,{\tilde{x}}_{N})$
is linearly stable.

Now suppose $\phi=1$, as similar to the case $\phi=0$, the fixed point is of
the form $(x_{1}^{*},x^{*}_{M},0)$, where $s_{M}f_{M}(x^{*}_{M})=1$ and
$x_{1}^{*}=\frac{x_{M}^{*}}{s_{M}}$. The equation $s_{M}f_{M}(x^{*}_{M})=1$
has two solution ${\tilde{x}_{M}}$ and ${\hat{x}_{M}}$ satisfying
${\tilde{x}_{M}}<C_{0}<{\hat{x}_{M}}$ which corresponds to two fixed points
${\tilde{x}}$ and ${\hat{x}}$ of the system (9). The eigen values of the
Jacobian, if we linearize the system near any fixed point
$x^{*}=(x_{1}^{*},x^{*}_{M},0)$ are
$\lambda=\pm\sqrt{1+s_{N}x^{*}_{M}f^{\prime}_{M}(x^{*}_{M})}$. So
$\lvert\lambda\rvert>1$ at $x^{*}={\tilde{x}}$ as
$f^{\prime}_{M}({\tilde{x}_{M}})>0$, hence it is an unstable fixed point and
$\lvert\lambda\rvert<1$ at $x^{*}={\hat{x}_{M}}$ by the same argument as in
the case $\phi=0$ so it is a stable fixed point.

So in summary, we just proved that for each $\phi\in[0,1]$, there is exactly
one locally stable fixed point for the system (1).

∎

##  3\. Evolutionary game and ESS

If Allee effect is present only in migrant population, the dynamics of the
system (1) undergoes a bifurcation as the zero fixed point stability changes
when parameter $\phi$ passes through $\phi_{1}$. The goal is to investigate
how the evolution of partial migration is affected when only the migrants
experience Allee effects. To address this question, we will use an
evolutionary Game theory approach as advocated in [9, 19, 20, 32]. In this
methodology, an individual’s allocation strategy is denoted by $v$, and the
mean allocation strategy $\phi(t)$ in the population is treated as a dynamic
state variable whose dynamics are governed by Lande’s equation (or the
breeder’s equation, Fisher’s equation, or the canonical equation of
evolution). The methodology provides a coupled system for the population
dynamics and the mean allocation strategy, known as the Darwinian dynamics:

(11) $\displaystyle\begin{pmatrix}x_{1}(t+1)\\\ x_{M}(t+1)\\\
x_{N}(t+1)\end{pmatrix}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0&f_{M}(z_{M}(t))&f_{N}(z_{N}(t))\\\
vs_{M}&0&0\\\
(1-v)s_{N}&0&0\end{pmatrix}\Biggr{\rvert}_{v=\phi(t)}\begin{pmatrix}x_{1}(t)\\\
x_{M}(t)\\\ x_{N}(t)\end{pmatrix}$ (12) $\displaystyle\phi(t+1)$
$\displaystyle=$
$\displaystyle\phi(t)+\sigma^{2}\frac{\partial\ln(\lambda(x(t),v)}{\partial
v}\Big{|}_{v=\phi(t)},$

Equation $(\ref{eg2})$ states that the change in the mean strategy is
proportional to the fitness gradient. Fitness here is taken to be
$\ln\left(\lambda(x,v)\right)$, where $\lambda(x,v)$ is the dominant
eigenvalue of the matrix

$A_{1}(x,v)=\begin{pmatrix}0&f_{M}(z_{M})&f_{N}(z_{N})\\\ vs_{M}&0&0\\\
(1-v)s_{N}&0&0\end{pmatrix}.$

The constant $\sigma^{2}$ is related to the (assumed constant) variance of the
strategy throughout the population (equal, or proportional to the variance,
depending on how the trait dynamics are derived) and is referred to as the
speed of evolution.

A straightforward calculation shows that $\lambda(x,v)$ equals the square root
of the basic reproduction number associated to $A_{1}(x,v)$, which we already
defined in $(\ref{R0:def})$:

$\lambda(x,v)=(R_{0}(x,v))^{1/2},\textrm{ where
}R_{0}(x,v):=vs_{M}f_{M}(z_{M})+(1-v)s_{N}f_{N}(z_{N}).$

Hence, system $(\ref{eg1})-(\ref{eg2})$ can be re-written as:

(13) $\displaystyle\begin{pmatrix}x_{1}(t+1)\\\ x_{M}(t+1)\\\
x_{N}(t+1)\end{pmatrix}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0&f_{M}(z_{M}(t))&f_{N}(z_{N}(t))\\\
\phi(t)s_{M}&0&0\\\
(1-\phi(t))s_{N}&0&0\end{pmatrix}\begin{pmatrix}x_{1}(t)\\\ x_{M}(t)\\\
x_{N}(t)\end{pmatrix}$ (14) $\displaystyle\phi(t+1)$ $\displaystyle=$
$\displaystyle\phi(t)+\frac{1}{2}\frac{\sigma^{2}}{R_{0}(x(t),\phi(t))}\frac{\partial
R_{0}(x(t),v)}{\partial v}\Big{|}_{v=\phi(t)},$

We first study this system for $\sigma=0$, i.e. when there are no evolutionary
forces at work:

######  Theorem 3.1.

Assume that $\sigma^{2}=0$. Suppose that $({\bf H1})-({\bf H3})$ hold,

Then the following holds:

  1. (1)

For every fixed $\phi_{0}$ in $(0,1)$, there is a positive fixed point
$(x^{*}(\phi_{0}),\phi_{0})$ such that

every positive solution of system $(\ref{eq13})-(\ref{eq14})$ with initial
condition $(x_{0},\phi_{0})$ for $x_{0}$ in a neighborhood of
$(x^{*}(\phi_{0})$, converges to $(x^{*}(\phi_{0}),\phi_{0})$. The fixed point
$(x^{*}(\phi_{0}),\phi_{0})$ is locally stable with respect to positive
initial conditions with arbitrary positive $x_{0}$, but fixed $\phi_{0}$.

  2. (2)

If $\phi_{0}=0$, then every positive solution of system
$(\ref{eq13})-(\ref{eq14})$ with initial condition $(x_{0},\phi_{0})$ for
arbitrary positive $x_{0}$, converges to a unique non-zero fixed point
$({\tilde{x}}_{1},0,{\tilde{x}}_{N},0)$, where ${\tilde{x}}_{N}>0$ is the
unique positive solution to the equation $s_{N}f_{N}(z)=1$, and
${\tilde{x}}_{1}={\tilde{x}}_{N}/s_{N}$. This fixed point is linearly stable
with respect to initial conditions with arbitrary positive $x_{0}$, but fixed
$\phi_{0}=0$.

  3. (3)

If $\phi_{0}=1$, then every positive solution of system
$(\ref{eq13})-(\ref{eq14})$ with initial condition $(x_{0},\phi_{0})$ for
arbitrary positive $x_{0}$, converges to a unique non-zero fixed point
$({\hat{x}}_{1},{\hat{x}}_{M},0,1)$, where ${\hat{x}}_{M}>0$ is a positive
solution to the equation $s_{M}f_{M}(z)=1$, and
${\hat{x}}_{1}={\hat{x}}_{M}/s_{M}$. This fixed point is linearly stable with
respect to initial conditions with arbitrary positive $x_{0}$, but fixed
$\phi_{0}=1$.

###### Proof.

The proof follows immediately from Theorem 2.10, and the fact that for each
$\phi_{0}$ in $[0,1]$, the set
$\\{(x,\phi)\in\mathbb{R}^{3}_{+}\times[0,1]\;|\;\phi=\phi_{0}\\}$ is forward
invariant for solutions of system $(\ref{eq13})-(\ref{eq14})$ when
$\sigma^{2}=0$. ∎

######  Definition 3.2.

Suppose $(x^{*}(\phi^{*}),\phi^{*})$ is the stable fixed point of the system
$\eqref{eq13}-\eqref{eq14}$ for $\phi^{*}$ in $[0,1]$ and for $\sigma=0$. We
say $\phi^{*}$ is an evolutionary stable strategy (ESS) if
$(x^{*}(\phi^{*}),\phi^{*})$ is a locally asymptotically stable fixed point of
system $(\ref{eq13})-(\ref{eq14})$ for small positive $\sigma^{2}\neq 0$.

This notion captures that if the population has adopted an ESS, then it can
not be invaded by other population that use nearby strategies. Our next result
shows the existence and uniqueness of an ESS.

######  Theorem 3.3.

Assume that $\sigma^{2}>0$. Suppose that $({\bf H1})-({\bf H3})$ hold, and the
carrying capacities satisfies the inequalities :
$q{\hat{x}_{M}}<{\tilde{x}_{N}}$ and $p{\tilde{x}_{N}}<{\hat{x}_{M}}$. Then
the system $(\ref{eq13})-(\ref{eq14})$ has a fixed point
$(x^{*}(\phi^{*}),\phi^{*})$ in $\mathbb{R}^{3}_{+}\times[0,1]$, where
$x^{*}(\phi^{*})$ is the unique stable positive fixed point of system
$(\ref{sys-1})$ with $\phi=\phi^{*}$ (see Theorem 2.10) and $\phi^{*}$ is
given by the following formula

(15)
$\phi^{*}=\frac{\frac{{\hat{x}}_{M}-p{\tilde{x}}_{N}}{{\tilde{x}}_{N}-q{\hat{x}}_{M}}}{\frac{{\hat{x}}_{M}-p{\tilde{x}}_{N}}{{\tilde{x}}_{N}-q{\hat{x}}_{M}}+\frac{s_{M}}{s_{N}}}.$

Moreover, $(x^{*}(\phi^{*}),\phi^{*})$ is a locally asymptotically stable
fixed point of system $(\ref{eq13})-(\ref{eq14})$ for sufficiently small
positive $\sigma^{2}.$ So $\phi^{*}$ is a unique ESS.

###### Proof.

The system of equations, when evolutionary force at work are

(16) $\displaystyle\begin{pmatrix}x_{1}(t+1)\\\ x_{M}(t+1)\\\
x_{N}(t+1)\end{pmatrix}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0&f_{M}(z_{M}(t))&f_{N}(z_{N}(t))\\\
\phi(t)s_{M}&0&0\\\
(1-\phi(t))s_{N}&0&0\end{pmatrix}\begin{pmatrix}x_{1}(t)\\\ x_{M}(t)\\\
x_{N}(t)\end{pmatrix}$ (17) $\displaystyle\phi(t+1)$ $\displaystyle=$
$\displaystyle\phi(t)+\frac{1}{2}\frac{\sigma^{2}}{R_{0}(x(t),\phi(t))}\frac{\partial
R_{0}(x(t),v)}{\partial v}\Big{|}_{v=\phi(t)},$

$\textrm{ where }R_{0}(x,v):=vs_{M}f_{M}(z_{M})+(1-v)s_{N}f_{N}(z_{N}).$
$\frac{\partial R_{0}(x(t),v)}{\partial
v}\Big{|}_{v=\phi(t)}=s_{M}f_{M}(z_{M})-s_{N}f_{N}(z_{N})$

Suppose $(x_{1}^{*},x_{M}^{*},x_{N}^{*},\phi^{*})$ is a fixed point for the
system (16)-(17). We have

$\phi^{*}(t)=\phi^{*}(t)+\frac{1}{2}\frac{\sigma^{2}(s_{M}f_{M}(z^{*}_{M})-s_{N}f_{N}(z^{*}_{N}))}{\phi^{*}s_{M}f_{M}(z^{*}_{M})+(1-\phi^{*})s_{N}f_{N}(z^{*}_{N})}$

which implies that $s_{M}f_{M}(z^{*}_{M})-s_{N}f_{N}(z^{*}_{N})=0$ so
$s_{M}f_{M}(z^{*}_{M})=s_{N}f_{N}(z^{*}_{N})$. We also have

$R_{0}(x^{*},\phi^{*})=\phi^{*}(s_{M}f_{M}(z^{*}_{M})-s_{N}f_{N}(z^{*}_{N}))+s_{N}f_{N}(z^{*}_{N})=1$

It follows that $s_{N}f_{N}(z^{*}_{N})=1$ and $s_{M}f_{M}(z^{*}_{M})=1$. The
equation $s_{N}f_{N}(z_{N})=1$ has unique solution say ${\tilde{x}}_{N}$ and
$s_{M}f_{M}(z_{M})=1$ has two solution ${\tilde{x}}_{M}$ and ${\hat{x}}_{M}$
with $f^{\prime}({\tilde{x}}_{M})>0$, $f^{\prime}({\hat{x}}_{M})<0$ and
${\tilde{x}}_{M}<{\hat{x}}_{M}.$ Hence
$\text{either}\;\>z^{*}_{M}(\phi)=x^{*}_{M}(\phi)+px^{*}_{N}(\phi)={\hat{x}}_{M}\;\>\text{or}\;\>z^{*}_{M}(\phi)={\hat{x}}_{M}\textrm{
and }z^{*}_{N}(\phi)=x^{*}_{N}(\phi)+qx^{*}_{M}(\phi)={\tilde{x}}_{N}$ After
simplifying and solving for $x_{M}^{*}$ and $x_{N}^{*}$ in case of
$z^{*}_{M}(\phi)={\hat{x}}_{M}$ we get

(18) $\displaystyle x^{*}_{M}(\phi)$ $\displaystyle=$
$\displaystyle\frac{{\hat{x}_{M}}-p{\tilde{x}_{N}}}{1-pq}$ (19) $\displaystyle
x^{*}_{N}(\phi)$ $\displaystyle=$
$\displaystyle\frac{{\tilde{x}_{N}}-q{\hat{x}_{M}}}{1-pq}$

it follows that:

(20)
$\frac{x^{*}_{M}(\phi)}{x^{*}_{N}(\phi)}=\frac{{\hat{x}_{M}}-p{\tilde{x}_{N}}}{{\tilde{x}_{N}}-q{\hat{x}_{M}}}=\frac{s_{M}}{s_{N}}\frac{\phi}{1-\phi}.$

(21)
$\phi^{*}=\frac{\frac{{\hat{x}}_{M}-p{\tilde{x}}_{N}}{{\tilde{x}}_{N}-q{\hat{x}}_{M}}}{\frac{{\hat{x}}_{M}-p{\tilde{x}}_{N}}{{\tilde{x}}_{N}-q{\hat{x}}_{M}}+\frac{s_{M}}{s_{N}}}.$

Similarly in case of $z^{*}_{M}(\phi)={\tilde{x}}_{M}$, we have

(22) $\displaystyle x^{*}_{M}(\phi)$ $\displaystyle=$
$\displaystyle\frac{{\tilde{x}_{M}}-p{\tilde{x}_{N}}}{1-pq}$ (23)
$\displaystyle x^{*}_{N}(\phi)$ $\displaystyle=$
$\displaystyle\frac{{\tilde{x}_{N}}-q{\tilde{x}_{M}}}{1-pq}$

Because $\tilde{x}_{M}<C_{0}<p\tilde{x}_{N}$, the corresponding fixed point is
not a positive one. So we have one positive fixed point for the system
(16)-(17), we will show that the fixed point $(x^{*}(\phi^{*}),\phi^{*})$ is
local asymptotically stable. We linearize the Darwinian system (16)-(17) near
fixed point $(x^{*}(\phi^{*}),\phi^{*})$ yielding the following Jacobian
matrix:
$J_{D}(\sigma^{2})=$ $\begin{pmatrix}0&a&b&0\\\ c&0&0&g\\\ d&0&0&-h\\\
0&e&f&1\end{pmatrix}$ where
$a=f_{M}^{\prime}(z_{M}^{*})x_{M}^{*}+f_{M}(z_{M}^{*})+x_{N}^{*}f^{\prime}_{N}(z_{N}^{*}).q$,
$b=pf_{M}^{\prime}(z_{M}^{*})x_{M}^{*}+f_{N}^{\prime}(z_{N}^{*})x_{N}^{*}+f_{N}(z_{N}^{*})$,
$c=\phi^{*}s_{M}$, $g=s_{M}x_{1}^{*}$ , $d=(1-\phi^{*})s_{N}$,
$h=s_{N}x_{1}^{*}$,
$e=\frac{\sigma^{2}}{2}(s_{M}f^{\prime}_{M}(z_{M}^{*})-qs_{N}f^{\prime}_{N}(z_{N}^{*}))$
and
$f=\frac{\sigma^{2}}{2}(ps_{M}f^{\prime}_{M}(z_{M}^{*})-s_{N}f^{\prime}_{N}(z_{N}^{*})).$

In the appendix, we have shown that the dominant eigenvalue of the matrix
$J_{D}(\sigma^{2})$ is positive and less than one, hence the $\phi^{*}$ given
by the equation $\eqref{IFD-value}$ is an unique ESS.

∎

##  4\. The ESS is an IFD in two habitat environment

Theorem 3.3 revels the existence of an evolutionary stable strategy
$\phi^{*}\in(0,1)$ such that for any population with the fixed strategy
$\phi^{*}$, can not be invaded by other population with near by strategies.
However, it does not say whether that strategy which lead to stable fixed
points that correspond to an ideal free distribution. The IFD has emerged in
many studies on the evolution of dispersal [3, 4, 8, 13, 15, 18, 25]. Here we
study the IFD concept for a population that exhibits partial migration. The
goal of this Section is to examine this issue. The concept of an IFD is based
on the ability of individuals to assess the quality of a spatial environment,
yet model (1) does not incorporate space explicitly. However, it does contain
space implicitly, and we can re-interpret model (1) as a population model that
evolves in two habitats $h_{M}$ and $h_{N}$, representing a migratory and non-
migratory habitat. To each habitat we can associate a pay-off or fitness
function which depends on the density of the abundances in that habitat, at
equilibrium state. A population at equilibrium is considered to be ideal free,
if the finesses in both habitats are equal and maximal. In this scenario,
individuals would have no incentive to move to a different habitat. In this
section, we will show that the ESS is in fact an IFD, corresponding to a non-
extreme value of the strategy parameter $\phi^{*}$. But first, we give a
mathematical definition of the IFD concept.

######  Definition 4.1.

Recall from Theorem 2.10 that for each $\phi\in[0,1]$, there is one positive
equilibrium state $(x^{*}_{1}(\phi),x_{M}^{*}(\phi),x_{N}^{*}(\phi))$ which is
locally asymptotically stable for system (1). We define the payoff (or
fitness) function for habitat $h_{I}$ to be
$R_{0}^{I}(z^{*}(\phi^{*}))=s_{I}f_{I}(z_{I}^{*})$, where $I=M,N$ ,
$z^{*}_{M}(\phi)=x^{*}_{M}(\phi)+px^{*}_{N}(\phi)$ and
$z^{*}_{N}(\phi)=x^{*}_{N}(\phi)+qx^{*}_{M}(\phi)$. An allocation strategy
$\phi^{*}$ is said to be an IFD strategy for model $\eqref{sys-1}$ if the two
habitats $h_{M}$ and $h_{N}$ have the same payoffs at $\phi^{*}$, i.e if

$R_{0}^{M}(z^{*}_{M}(\phi^{*}))=R_{0}^{N}(z^{*}_{N}(\phi^{*})),$

and if no other strategy satisfying this payoff equality condition, has a
higher payoff.

######  Theorem 4.2.

If the conditions of Theorem 3.3 hold for system (1) then the ESS strategy
$\phi^{*}$ given by the formula (15) is the only IFD.

###### Proof.

By formula (15), we have the following;

$\frac{{\hat{x}_{M}}-p{\tilde{x}_{N}}}{{\tilde{x}_{N}}-q{\hat{x}_{M}}}=\frac{s_{M}}{s_{N}}\frac{\phi^{*}}{1-\phi^{*}}.$

Choose $\tilde{x}_{1}=\frac{{\hat{x}_{M}}-p{\tilde{x}_{N}}}{(1-pq)\phi
s_{M}}=\frac{{\tilde{x}_{N}}-q{\hat{x}_{M}}}{(1-pq)(1-\phi)s_{N}}$ and
${\tilde{x}}=({\tilde{x}_{1}},\frac{{\hat{x}_{M}}-p{\tilde{x}_{N}}}{1-pq},\frac{{\tilde{x}_{N}}-q{\hat{x}_{M}}}{1-pq})$.
It is easy to check that $R_{0}({\tilde{x}},\phi^{*})=1$ so, clearly
${\tilde{x}}$ is an equilibrium point of the system (1) which is positive. One
can show that it is in fact locally asymptotically stable. So we have

$x^{*}_{M}(\phi^{*})=\frac{{\hat{x}_{M}}-p{\tilde{x}_{N}}}{1-pq}\textrm{ and
}x^{*}_{N}(\phi^{*})=\frac{{\tilde{x}_{N}}-q{\hat{x}_{M}}}{1-pq}$

Hence the pay-off functions are found to be equal:

$R_{0}^{M}(z^{*}_{M}(\phi^{*}))=R_{0}^{M}({\hat{x}_{M}})=1=R_{0}^{N}({\tilde{x}_{N}})=R_{0}^{N}(z^{*}_{N}(\phi^{*}))$

and this proves that $\phi^{*}$ is indeed an IFD.

Conversely suppose $\phi=\phi^{*}$ is an IFD. Then

$R_{0}^{M}(z^{*}_{M}(\phi^{*}))=R_{0}^{N}(z^{*}_{N}(\phi^{*}))$

By the theorem (3.3), $\phi^{*}$ is an ESS. ∎

##  5\. discussion

The goals of this study were twofold: investigating if the evolution of
partial migration is affected when only the migrants experience Allee effects
and using the evolutionary game theory approach to determine evolutionary
stable strategies if any. In existing literature [14, 21, 24, 29], it is shown
that negative density dependence in the fertilities alone can explain the
partial migration phenomenon, provided it is attenuated with increasing sub-
type abundances. The works are mostly numerical. Nevertheless, several
biological features specially Allee effects of partially migrating populations
have been neglected. An Allee effect is a positive association between
absolute average individual fitness and population size over some finite
interval. In some partially migrating population, the migrant individual
experience predation during their stays at migrant habitat which may result in
Allee effects. When the size of populations subject to Allee effects is low,
then these populations tend towards extinction. This fact argues for a
thorough understanding of Allee effects and their mechanisms in order to
develop sound management practices for a number of environmental issues. So
the important aspect of this work is that, we have obtained an analytical
formula for the Evolutionary Stable Strategy (ESS) for the allocation strategy
adopted by a partially migrating population with migrant population
experiencing a strong Allee effects. This ESS is expressed in terms of the
demographic model parameters for the migrant and non-migrant populations, and
thus the formula can be used to predict the ESS value, whenever the life
histories of the migrant and non-migrant populations are known, for example
from lab or empirical data. More ever, the conditions on environmental
parameters like carrying capacities under which the population will be
evolutionary stable are clearly stated. Our results differ from previous
studies [14, 21, 24, 29] in that partial migration as it point to more of a
system of thresholds, which provides some insight into how populations may
respond to future conditions when migrant undergoes Allee effects. We also
investigate the connection between spatio/temporal structure and the evolution
of partial migration behavior by associating the ESS to Ideal Free
distribution (IFD) which happens to be a powerful tool for understanding how
populations distribute themselves in heterogeneous environments. Mostly the
phenomenon is studied in temporally constant environment but many environments
are seasonal and changes with time. So for the future work it is motivating to
investigate the phenomenon when the underlying population model is modified to
incorporate time varying environments.

##  6\. Acknowledgment

This work is funded by start up research grant, SERB (SRG/2019/002200) to AM.

##  7\. Appendix

The dominant eigen value of the Jacobian of Darwinian system:

The Jacobian matrix with $\sigma^{2}>0$ is given by

$J_{D}(\sigma^{2})=\begin{pmatrix}C(x^{*}(\phi^{*}),\phi^{*})&{\bf*}\\\
0&1\end{pmatrix}+\sigma^{2}\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&\frac{(s_{M}f^{\prime}_{M}(z_{M}^{*})-qs_{N}f^{\prime}_{N}(z_{N}^{*}))}{2}&\frac{(ps_{M}f^{\prime}_{M}(z_{M}^{*})-s_{N}f^{\prime}_{N}(z_{N}^{*}))}{2}&0\end{pmatrix},$

where the ${\bf*}$ represents a $3$-dimensional vector whose value is
unimportant at present, and the $3\times 3$ matrix
$C(x^{*}(\phi^{*}),\phi^{*})$ is defined as:

$C(x^{*}(\phi^{*}),\phi^{*})=\begin{pmatrix}0&a&b\\\ c&0&0\\\
d&0&0\end{pmatrix},$

with
$a=f_{M}^{\prime}(z_{M}^{*})x_{M}^{*}+f_{M}(z_{M}^{*})+x_{N}^{*}f^{\prime}_{N}(z_{N}^{*}).q,b=pf_{M}^{\prime}(z_{M}^{*})x_{M}^{*}+f_{N}(z_{M}^{*})+x_{N}^{*}f^{\prime}_{N}(z_{N}^{*}),c=\phi^{*}s_{M}$
and $d=(1-\phi^{*})s_{N}$. One can check that the eigen values of the matrix
$C(x^{*}(\phi^{*}),\phi^{*})$ are $0$ and
$\pm\sqrt{1+x_{1}^{*}Q^{{}^{\prime}}(x_{1}^{*})}$. We have proved in the
theorem (2.10) that the dominant eigen value
$|\sqrt{1+x_{1}^{*}Q^{{}^{\prime}}(x_{1}^{*})}|$ of the matrix
$C(x^{*}(\phi^{*}),\phi^{*})$ is positive and less than one. Consequently, the
dominant eigenvalue of $J_{D}(0)$ is $1$, and this eigenvalue is simple. By
continuity of eigenvalues, the matrix $J_{D}(\sigma^{2})$ will also have a
real, simple and dominant eigenvalue $\lambda_{p}(\sigma^{2})$ for all
sufficiently small $\sigma^{2}$, such that $\lambda_{p}(0)=1$. We claim that
$\lambda_{p}(\sigma^{2})<1$, at least for all sufficiently small $\sigma^{2}$.
To prove this, we now examine the roots of the characteristic polynomial
$F(\lambda,\sigma^{2}):=\det(\lambda I-J_{D}(\sigma^{2}))$ associated to the
matrix $J_{D}(\sigma^{2})$.

(24) $J_{D}(\sigma^{2})=\begin{pmatrix}0&a&b&0\\\ c&0&0&g\\\ d&0&0&-h\\\
0&\sigma^{2}e&\sigma^{2}f&1\end{pmatrix}$

Where $g=s_{M}x_{1}^{*}$ , $h=s_{N}x_{1}^{*}$,
$e=\frac{\sigma^{2}}{2}(s_{M}f^{\prime}_{M}(z_{M}^{*})-qs_{N}f^{\prime}_{N}(z_{N}^{*}))$
and
$f=\frac{\sigma^{2}}{2}(ps_{M}f^{\prime}_{M}(z_{M}^{*})-s_{N}f^{\prime}_{N}(z_{N}^{*})).$
A tedious calculation shows that the characteristic polynomial of the matrix
$J_{D}(\sigma^{2})$ is given by:

$F(\lambda,\sigma^{2})=\lambda^{4}-\lambda^{3}-\left[\sigma^{2}(eg-
fh)+(ac+bd)\right]\lambda^{2}+(ac+bd)\lambda+\sigma^{2}(be-af)(ch+dg)$

Note that $F(\lambda,0)$ is positive for all $\lambda>1=\lambda_{p}(0)$ (since
$\lambda_{p}(0)=1$ is the dominant root of $F(\lambda,0)$, and
$\lim_{\lambda\to\infty}F(\lambda,0)=+\infty$). Moreover, $\partial
F/\partial\lambda(\lambda_{p}(0),0)$ must be positive because
$\lambda_{p}(0)=1$ is a simple root of $F(\lambda,0)$ (this can also be shown
directly by calculating this partial derivative using the expression above:
$\partial F/\partial\lambda(\lambda_{p}(0),0)=1-(ac+bd)$, and this is
positive). Therefore, the claim above (namely, that
$\lambda_{p}(\sigma^{2})<1$, for all sufficiently small $\sigma^{2}$) will be
proved, provided we can show that for all sufficiently small $\sigma^{2}$,
there holds that:

(25) $F(1,\sigma^{2})>0.$

Evaluating $F(1,\sigma^{2})$ yields:

$F(1,\sigma^{2})=\sigma^{2}\left((be-af)(ch+dg)-(eg-fh)\right).$

Therefore, a sufficient condition for $(\ref{stable})$ to hold, is that:

$(fh-eg)-((af-be)(ch+dg))>0$

After simplifying we found that

$(fh-eg)-((af-
be)(ch+dg))=f_{M}^{\prime}(z^{*}_{M})f^{\prime}_{N}(z^{*}_{N})s_{M}s_{N}(1-pq)(x^{*}_{N}s_{M}+x^{*}_{M}s_{N})>0$

This concludes that the dominant eigen value of the Jacobian matrix
corresponding to the Darwinan system (16)-(17) is less than one in modulus.

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# Nonextensive Effect on the Lump Soliton Structures in Dusty Plasma

Prasanta Chatterjee  Uday Narayan Ghosh  Department of Mathematics, KKM
College, A Constituent unit of Munger, Bihar-811307, India  Snehalata Nasipuri
M.Ruhul Amin  Department of Mathematical and Physical Sciences, East West
University, Aftabnagar, Dhaka-1212

###### Abstract

In this paper, we use a very prominent technique, Hirota Bilinear Method (HBM)
to survey the lump structures of the Kadomtsev-Petviashvili (KP) equation in
the frame of a collisionless magnetized plasma system composed of dust grains,
ions, and nonextensive electrons. Nonlinearity has worldwide applications, and
soliton theory is a powerful appliance to illustrate its qualitative
behaviors. So, lump solitons are very significant and also interesting. We
have observed that lump structures differ due to the correlated parameters of
the plasma system. It has also been found that the nonextensive parameter
crucially changes the lump features.

KEY WORDS: Lump solitons; Hirota Bilinear Method; KP equation; Nonextensive
electrons; Dusty Plasma.

##  1 Introduction

In advanced research in plasma physics, dusty plasma is a weighty and
captivating research area. It is located in a major part of the space plasma
and solar system. Due to its majority in space plasma [1]-[2] and the solar
system, it holds an enlarged portion of plasma research. It covers a broad
area of space plasma and the solar system. So it plays a crucial role in the
plasma physics research field. Dusty plasma becomes a hot research topic for
its enormous exposition and some remarkable properties [3]-[5]. Dust grains
are highly charged in the ionized medium. Dust grains are highly inactive and
charge-centralized, so the united characteristics of the electromagnetic force
field, which can authorize the circulation of ion-acoustic solitary waves
(IASWs), can be reformed by the charged dust grains. Subsequently, the dusty
plasma produced some low-frequency collective states, which are remarkably
different from the typical electron-ion plasmas. Experimental and theoretical
examinations [6]-[10] confess the creation of low-frequency and highly low-
frequency acoustic states, which have practical uses in space plasma along
with laboratory plasma.

The interaction of materiality and recovering force generates various states.
While the electron-ion pressure originates the recovering force, the
materiality is raised by the excessive dust grains. It was first theoretically
introduced by Rao et al. [6] in unmagnetized dusty plasma that an ion-acoustic
wave (IAW) is in such a variety of states. Dust ion-acoustic solitary waves
(DIASWs) were predicted by Shukla and Silin [11] and Barkan et al. [12] and
Melrino et al. [13] experimentally observed them in the laboratory. The dust
acoustic shock waves [14] were produced by the laboratory examinations of Iowa
State University (USA) [15]and the Institute of Space and Astronautical
Science (Japan) [16] .

To describe the nonlinear structures, chaos theory and soliton theory are two
efficient branches. Solitons consist of the most important properties of
particles as well as waves, which reflect the nonlinear structures in a
disciplined way. So the study of soliton solutions of nonlinear evolution
equations is undeniably required. Solitons propagate with a constant
configuration. Even after interaction with another soliton of the same kind,
they preserved their shape, velocity, and amplitude [17]-[20]. Some famous
nonlinear partial differential equations, like Korteweg-de Vries (KdV)
equation, Kadomtsev-Petviashvili (KP) equation, and Zakharov–Kuznetsov (ZK)
equation, proceed with soliton solutions. Extended and modified direct
algebraic method, extended mapping method, trigonometric function method, the
inverse scattering transformation [21], [22] Backlund transformation, Hirota
bilinear method [23], and Darboux transformation method [24] are very useful
due to their efficiency in getting isolated solutions.
Usually solitons have elastic nature but some are non-elastic, like kink
solitons, waves from Burger’s equation [25], rational breather waves, kinky
waves [26] etc. A special type of soliton, which is a rational function
solution localized in all directions of space, is known as a lump soliton
[17]\- [18]. It is a large-amplitude solitary wave, having similarities with
the rough wave. Its occurrence is uncertain and vanishes without any hint.
Such interesting nonlinear features are much more attractive, and the physical
significance has drawn the attention of many researchers. The power of
offensiveness, the devastation of sturdy nature, and the utility of signal
exhibition in optical fibers, plasma physics, laser and optical physics, gas
dynamics, hydrodynamics, and electromagnetics [27] make the study of lump
solitons impressive. There are many integrable nonlinear equations, like the
KdV equation, the Davey-Stewartson-II equation [28], the three-dimensional
three-wave resonant interaction equation [29], the BKP equation [30], the
Kadomtsev–Petviashvili-I (KP-I) equation [31], those produces lump solutions
[32]-[35]. Lump structures exist in many integrable equations [36]-[37]. Ma et
al. [38] has derived a class of lump solutions from such an integrable
equation named the KP equation using the HBM. The expansive occurrence of lump
solitons in the nonlinear evolution equations, especially the KP equation,
impresses the researchers [39]-[42] enough to expose their excitable nonlinear
interpretation.

Using transverse perturbations while Boris Kadomtsev and Vladimir Petviashvili
[43] were investigating the one-soliton solution stability of the KdV
equation, they deduced the KP equation. The KP equation is a renowned model
equation in nonlinear wave theory. The nonlinearity and dispersion created by
the ambiguity balanced each other in the KP equation. When the consequences of
transverse direction are merged, the extended KdV equation in 2-dimensional
space is converted to the KP equation. The studies [44]-[48] of nonlinearities
in various plasma fields through the KP equation have become more interesting.
Kumar and Malik [49] have examined the propagation of soliton and also noted
the presence of compressive soliton only per the KP equation in magnetized
dusty plasma. Utilizing a quantum hydrodynamical model in an electron-ion
Fermi plasma, Mushtaq et al. [50] investigated magneto-acoustic waves through
the soliton solution of the KP equation with quantum diffraction effects.
Masood et al. [51] analysed the low-frequency features transversely
propagating magneto-acoustic waves in the KP model in condensed electron-
positron-ion (e-p-i) magnetoplasma.

The above discussions concluded that, the lump soliton solutions are well-
considered in various fields in nonlinear sciences, theoretical physics, and
water wave theory, yet no further investigation has been performed in plasma
science. Inspired by these inquiries and fascinated by the inherent
characteristics of lump structures, we have investigated the existence of lump
solitons and their significance in parameter-dependent systems.

##  2 Governing Equations

We consider the basic hydrodynamical model Eqs., as follows

$\frac{\partial n_{i}}{\partial T}+\nabla^{\prime}.(n_{i}v_{i})=0$ (2.1)
$\frac{\partial v_{i}}{\partial
T}+(v_{i}.\nabla^{\prime})v_{i}=-\frac{e\nabla^{\prime}\psi}{m_{i}}+\frac{eB_{0}}{m_{i}c}v_{i}\times
e_{z}$ (2.2) $(\nabla^{\prime})^{2}\psi=-4\pi[-en_{e}+en_{i}-ez_{d}n_{d}]$
(2.3)

$n_{i}$ and $v_{i}$ are the number density and velocity of ions, $m_{i}$ is
the mass of ions, $n_{e}$ and $n_{d}$ are the number densities of electrons
and dust respectively. $\psi$ is the electrostatic potential and
$q_{d}=-ez_{d}$ is the dust charge, where $z_{d}$ is the dust charge number
and $e$ is the elementary charge of electrons. Now we normalized the basic
variables in Eqs.(2.1)-(2.3) as follows

$t=\Omega
T,\nabla=\frac{c_{s}}{\Omega}\nabla^{\prime},v=\frac{v_{i}}{c_{s}},n=\frac{n_{i}}{n_{i0}},\phi=\frac{e\psi}{T_{e}}$

Different power-law functions have been established due to the necessity of
dynamics of distinct plasma situations in uniform magnetic plasma field. The
power law tails are originated obviously by the effect of collision cross-
section and mean free path on particle velocity and also the particle
trajectories are changed amazingly. The difference between the collision
cross-section and particle speed means diverges the tail of the velocity
distribution from Gaussian/Maxwellian distribution. Space plasmas, plasmas in
planetary magnetospheres, solar wind, magneto sheaths of magnetized planets,
ionosphere, and planetary magnetospheres, [52]-[54] as well as astrophysical
plasmas, are distinguished more generally by an appropriate velocity
distribution function, called q nonextensive distribution [55] and is
recognized by Renyi [56], where the entropic index q, is the degree of non
extensively of the considered system. Due to the wide range covering
capability of plasma situation, nonextensive distribution is rather admissible
than the general Maxwellian distributio. Many researchers [57]-[62] have
observed nonlinear structures including non-Maxwellian electrons distribution
function. When the electrons obey non-extensive Tsallis distribution, the
density of electrons is given by

$n_{e}=\left[1+(q-1)\phi\right]^{\frac{(q+1)}{2(q-1)}}$ (2.4)

$\implies$

$n_{e}=1+a_{1}\phi+b_{1}\phi^{2}+c_{1}\phi^{3}+\dots$ (2.5)

where $a_{1}=\frac{(q+1)}{2}$ , $b_{1}=\frac{(q+1)(3-q)}{8}$ ,
$c_{1}=\frac{(q+1)(3-q)(5-3q)}{48}$
where $\lambda_{e}={\left(\frac{T_{e}}{4\pi n_{e0}e^{2}}\right)}^{1/2}$ is the
electron Debye length, $c_{s}={\left(\frac{T_{e}}{m_{i}}\right)}^{1/2}$ is the
ion acoustic velocity, $\Omega=\frac{eB_{0}}{m_{i}c}$ is the ion
gyrofrequency. $n_{e0}$, $n_{i0}$ are the electron and ion densities
respectively in the unperturbed state.

##  3 Normalized Equations

Using the Eqs.(2.1)-(2.5), we get the normalized Eqs. as

$\frac{\partial n}{\partial t}+\nabla.(nv)=0$ (3.1) $\frac{\partial
v}{\partial t}+(v.\nabla).v=-\nabla\phi+(v\times e_{z})$ (3.2)
$\nabla^{2}\phi=\beta[(1+a_{1}\phi+b_{1}\phi^{2}+c_{1}\phi^{3}+\dots)-\delta_{1}n+\delta_{2}]$
(3.3)

Where $\beta=\frac{4\pi e^{2}}{T_{e}}{r_{g}^{2}}$, $\delta_{1}$ and
$\delta_{2}$ are the ratio of ions and dust grains to the electrons number
density and $r_{g}=\frac{c_{s}}{\Omega}$ is the ion gyroradius.

##  4 Derivation of Kadomtsev-Petviashvili (KP) Equation

We assume that the wave is propagating in the x-z direction. Then the
normalized Eqs. reduce to

$\frac{\partial n}{\partial t}+\frac{\partial(nv_{x})}{\partial
x}+\frac{\partial(nv_{z})}{\partial z}=0$ (4.1) $\frac{\partial
v_{x}}{\partial t}+\left(v_{x}\frac{\partial}{\partial
x}+v_{z}\frac{\partial}{\partial z}\right)v_{x}=-\frac{\partial\phi}{\partial
x}+v_{y}$ (4.2) $\frac{\partial v_{y}}{\partial
t}+\left(v_{x}\frac{\partial}{\partial x}+v_{z}\frac{\partial}{\partial
z}\right)v_{y}=-v_{x}$ (4.3) $\frac{\partial v_{z}}{\partial
t}+\left(v_{x}\frac{\partial}{\partial x}+v_{z}\frac{\partial}{\partial
z}\right)v_{z}=-\frac{\partial\phi}{\partial z}$ (4.4)
$\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial
z^{2}}\right)\phi=\beta[(1+a_{1}\phi+b_{1}\phi^{2}+c_{1}\phi^{3}+\dots)-\delta_{1}n+\delta_{2}]$
(4.5)

To linearize the normalized Eqs. (4.1)-(4.5), let us write the dependent
variables as the sum of equilibrium and perturbed parts, so, $n=1+\bar{n}$,
$v_{x}=\bar{v_{x}}$, $v_{y}=\bar{v_{y}}$, $v_{z}=\bar{v_{z}}$ and
$\phi=\bar{\phi}$. After linearizing, we take all perturbed variables of the
form $e^{i(k_{x}x+k_{z}z-\omega t)}$, where $k_{x}$, $k_{z}$ are the wave
numbers in $x$ and $z$ direction respectively, and $\omega$ is the wave
frequency $(\omega<<\Omega)$. This leads to

$\begin{vmatrix}-i\omega&ik_{x}&0&ik_{z}&0\\\ 0&-i\omega&-1&0&ik_{x}\\\
0&1&-i\omega&0&0\\\ 0&0&0&-i\omega&ik_{z}\\\
\beta\delta_{1}&0&0&0&-(k^{2}_{x}+k^{2}_{z}+a_{1}\beta)\end{vmatrix}=0$ (4.6)

It gives the dispersion relation

$\omega=k_{z}\left[\frac{a_{1}}{\delta_{1}}+(1+\frac{1}{\beta\delta_{1}})k^{2}_{x}+(1+\frac{1}{\beta\delta_{1}})k^{2}_{z}\right]^{-1/2}$
(4.7)

###  4.1 Stretched Co-ordinates and Perturbation

The dispersion relation guides us which stretching should be adopted for the
fluid hydrodynamical model to obtain nonlinear evolution equations. On the
basis of the dispersion relation, we take the stretching co-ordinates as
follows

$\chi=\epsilon^{2}x$ (4.8) $\xi=\epsilon(z-Vt)$ (4.9) $\tau=\epsilon^{3}t$
(4.10)

where $V$ is the phase velocity of IAW and $\epsilon$ is a small parameter
measuring the strength of the non-linearity.
The dependent variables are expanded as

$n=1+\epsilon^{2}n_{1}+\epsilon^{4}n_{2}+\dots$ (4.11)
$v_{x}=\epsilon^{3}v_{x_{1}}+\epsilon^{5}v_{x_{2}}+\dots$ (4.12)
$v_{y}=\epsilon^{3}v_{y_{1}}+\epsilon^{5}v_{y_{2}}+\dots$ (4.13)
$v_{z}=\epsilon^{2}v_{z_{1}}+\epsilon^{4}v_{z_{2}}+\dots$ (4.14)
$\phi=\epsilon^{2}\phi_{1}+\epsilon^{4}\phi_{2}+\dots$ (4.15)

Then the normalized Eqs. (4.1)-(4.5) can be written in terms of $\chi$, $\xi$
and $\tau$ as follows

$-\epsilon V\frac{\partial n}{\partial\xi}+\epsilon^{3}\frac{\partial
n}{\partial\tau}+\epsilon^{2}\frac{\partial(nv_{x})}{\partial\chi}+\epsilon\frac{\partial(nv_{z})}{\partial\xi}=0$
(4.16) $-\epsilon V\frac{\partial
v_{x}}{\partial\xi}+\epsilon^{3}\frac{\partial
v_{x}}{\partial\tau}+\left(\epsilon^{2}v_{x}\frac{\partial
v_{x}}{\partial\chi}+\epsilon v_{z}\frac{\partial
v_{x}}{\partial\xi}\right)=-\epsilon^{2}\frac{\partial\phi}{\partial\chi}+v_{y}$
(4.17) $-\epsilon V\frac{\partial
v_{y}}{\partial\xi}+\epsilon^{3}\frac{\partial
v_{y}}{\partial\tau}+\left(\epsilon^{2}v_{x}\frac{\partial
v_{y}}{\partial\chi}+\epsilon v_{z}\frac{\partial
v_{y}}{\partial\xi}\right)=-v_{x}$ (4.18) $-\epsilon V\frac{\partial
v_{z}}{\partial\xi}+\epsilon^{3}\frac{\partial
v_{z}}{\partial\tau}+\left(\epsilon^{2}v_{x}\frac{\partial
v_{z}}{\partial\chi}+\epsilon v_{z}\frac{\partial
v_{z}}{\partial\xi}\right)=-\epsilon\frac{\partial\phi}{\partial\xi}$ (4.19)
$\left(\epsilon^{4}\frac{\partial^{2}}{\partial\chi^{2}}+\epsilon^{2}\frac{\partial^{2}}{\partial\xi^{2}}\right)\phi=\beta[(1+a_{1}\phi+b_{1}\phi^{2}+c_{1}\phi^{3}+\dots)-\delta_{1}n+\delta_{2}]$
(4.20)

Substituting (4.11)-(4.15) into the Eqs. (4.16)-(4.20) and equating the
coefficients of each power of $\epsilon$ to zero, we get the first order and
second order density, velocity and potential function. After some
simplifications, we obtain

$\frac{\partial}{\partial\xi}\left[\frac{\partial\phi_{1}}{\partial\tau}+A\phi_{1}\frac{\partial\phi_{1}}{\partial\xi}+B\frac{\partial^{3}\phi_{1}}{\partial\xi^{3}}\right]-C\frac{\partial^{2}\phi_{1}}{\partial\chi^{2}}=0$
(4.21)

where $A=\frac{-2b_{1}V^{2}+3a_{1}}{2a_{1}V}$, $B=\frac{V}{2a_{1}\beta}$ and
$C=\frac{V}{2}$ and $a_{1}=\frac{(1+q)}{2}$ , $b_{1}=\frac{(1+q)(3-q)}{8}$
$V^{2}=\frac{\delta_{1}}{a_{1}}$ is the phase velocity. Eq. (4.21) is known as
the KP equation.

###  4.2 Standard form of the KP Equation

We now take the alteration, $\xi$ by $\bar{\xi}A^{m_{1}}B^{n_{1}}C^{p_{1}}$,
$\phi_{1}$ by $6\bar{\phi_{1}}A^{m_{2}}B^{n_{2}}C^{p_{2}}$, $\chi$ by
$\frac{1}{\sqrt{\sigma}}\bar{\chi}A^{m_{3}}B^{n_{3}}C^{p_{3}}$ and $\tau$ by
$\bar{\tau}$.
Then we get

$m_{1}=0,m_{2}=-1,m_{3}=0$

$n_{1}=\frac{1}{3},n_{2}=\frac{1}{3},n_{3}=\frac{1}{6}$

$p_{1}=0,p_{2}=0,p_{3}=\frac{1}{2}$

$\implies$

$\xi=\bar{\xi}B^{\frac{1}{3}},\phi_{1}=6\bar{\phi_{1}}A^{-1}B^{\frac{1}{3}},\chi=\frac{1}{\sqrt{\sigma}}\bar{\chi}B^{\frac{1}{6}}C^{\frac{1}{2}},\tau=\bar{\tau}$
(4.22)

So, the Eq. (4.21) converted to

$\frac{\partial}{\partial\bar{\xi}}\biggr{[}\frac{\partial\bar{\phi_{1}}}{\partial\bar{\tau}}+6\bar{\phi_{1}}\frac{\partial\bar{\phi_{1}}}{\partial\bar{\xi}}+\frac{\partial^{3}\bar{\phi_{1}}}{\partial{\bar{\xi}}^{3}}\biggr{]}-\sigma\frac{\partial^{2}\bar{\phi_{1}}}{\partial{\bar{\chi}}^{2}}=0$

Omitting bar and subscript, we obtain the standard KP equation as

$\frac{\partial}{\partial\xi}\biggr{[}\frac{\partial\phi}{\partial\tau}+6\phi\frac{\partial\phi}{\partial\xi}+\frac{\partial^{3}\phi}{\partial{\xi}^{3}}\biggr{]}-\sigma\frac{\partial^{2}\phi}{\partial\chi^{2}}=0$
(4.23)

If $\sigma=1$, then Eq. (4.23) is called KP-I equation and when $\sigma=-1$,
Eq. (4.23) is called KP-II equation.

##  5 Lump Solutions of the KP Equation

In this section, we will derive the lump solutions for KP-I equation.
For this, we consider a transformation such that the dependent variable
$\phi(\xi,\chi,\tau)$ converted to a new dependent variable
$f(\xi,\chi,\tau)$, as follows

$\begin{split}\phi(\xi,\chi,\tau)=&2\frac{\partial^{2}}{\partial\xi^{2}}\left(lnf\right)\\\
&=2\left[\frac{f_{\xi\xi}f-f^{2}_{\xi}}{f^{2}}\right]\end{split}$ (5.1)

Then the KP-I equation ($\sigma=1$ in (4.23)), converted to the following
equation

$\left(f_{\xi\tau}f-f_{\xi}f_{\tau}\right)-\left(f_{\chi\chi}f-f^{2}_{\chi}\right)+f_{\xi\xi\xi\xi}f-4f_{\xi\xi\xi}f_{\xi}+3f^{2}_{\xi\xi}=0$
(5.2)

Now using Hirota derivatives, we obtain the Hirota bilinear form of Eq. (5.2)
as

$\left(D_{\xi}D_{\tau}+D^{4}_{\xi}\right)(f.f)-D^{2}_{\chi}(f.f)=0$ (5.3)

To derive lump solutions of KP-I equation, we consider the introduced
dependent variable $f$ as follows

$f=g^{2}+h^{2}+d_{9},\qquad g=d_{1}\xi+d_{2}\chi+d_{3}\tau+d_{4},\qquad
h=d_{5}\xi+d_{6}\chi+d_{7}\tau+d_{8},$ (5.4)

where $d_{i},1\leq i\leq 9$, are the real parameter.
Using the value of $f$, we get the following set of parameters

$\left\\{\begin{split}&d_{1}=d_{1},d_{2}=d_{2},d_{3}=\frac{d_{1}d^{2}_{2}-d_{1}d^{2}_{6}+2d_{2}d_{5}d_{6}}{d^{2}_{1}+d^{2}_{5}},\\\
&d_{4}=d_{4},d_{5}=d_{5},d_{6}=d_{6},d_{7}=\frac{2d_{1}d_{2}d_{6}-d^{2}_{2}d_{5}+d_{5}d^{2}_{6}}{d^{2}_{1}+d^{2}_{5}},\\\
&d_{8}=d_{8},d_{9}=\frac{3(d^{2}_{1}+d^{2}_{5})^{3}}{(d_{1}d_{6}-_{2}d_{5})^{2}}\end{split}\right\\}$
(5.5)

with the determinant condition

$\Delta:=d_{1}d_{6}-d_{2}d_{5}=\begin{vmatrix}d_{1}&d_{2}\\\
d_{5}&d_{6}\end{vmatrix}\neq 0$ (5.6)

Hence the introduced dependent variable expressed as

$\begin{split}f=&\left[d_{1}\xi+d_{2}\chi+\frac{d_{1}d^{2}_{2}-d_{1}d^{2}_{6}+2d_{2}d_{5}d_{6}}{d^{2}_{1}+d^{2}_{5}}\tau+d_{4}\right]^{2}\\\
&+\left[d_{5}\xi+d_{6}\chi+\frac{2d_{1}d_{2}d_{6}-d^{2}_{2}d_{5}+d_{5}d^{2}_{6}}{d^{2}_{1}+d^{2}_{5}}\tau+d_{8}\right]^{2}\\\
&+\frac{3(d^{2}_{1}+d^{2}_{5})^{3}}{(d_{1}d_{6}-d_{2}d_{5})^{2}},\end{split}$
(5.7)

and the lump solutions of KP-I equation are given by

$\phi(\xi,\chi,\tau)=\frac{4(d^{2}_{1}+d^{2}_{5})f-8(d_{1}g+d_{5}h)^{2}}{f^{2}}$
(5.8)

where the expression of $f$ is given in Eq. (5.7), and $g$ and $h$ are as
follows

$g=d_{1}\xi+d_{2}\chi+\frac{d_{1}d^{2}_{2}-d_{1}d^{2}_{6}+2d_{2}d_{5}d_{6}}{d^{2}_{1}+d^{2}_{5}}\tau+d_{4},$
(5.9)
$h=d_{5}\xi+d_{6}\chi+\frac{2d_{1}d_{2}d_{6}-d^{2}_{2}d_{5}+d_{5}d^{2}_{6}}{d^{2}_{1}+d^{2}_{5}}\tau+d_{8}.$
(5.10)

Now the lump solutions of the Eq. (4.21), as follows

$\phi(\xi,\chi,\tau)=4(\frac{6B^{1/3}}{A})\left[\frac{(d_{1}^{2}+d_{5}^{2})f-2(d_{1}g+d_{5}h)^{2}}{f^{2}}\right]$
(5.11)

where $g$ , $h$ and $f$ are given by

$g=\frac{1}{B^{1/3}C^{1/2}}\left[C^{1/2}d_{1}\xi+B^{1/6}d_{2}\chi+B^{1/3}C^{1/2}d_{3}\tau+B^{1/3}C^{1/2}d_{4}\right]$
(5.12)
$h=\frac{1}{B^{1/3}C^{1/2}}\left[C^{1/2}d_{5}\xi+B^{1/6}d_{6}\chi+B^{1/3}C^{1/2}d_{7}\tau+B^{1/3}C^{1/2}d_{8}\right]$
(5.13)

and

$\begin{split}f=&\frac{1}{B^{2/3}C}\Bigg{(}\left(C^{1/2}d_{1}\xi+B^{1/6}d_{2}\chi+B^{1/3}C^{1/2}d_{3}\tau+B^{1/3}C^{1/2}d_{4}\right)^{2}\\\
&+\left(C^{1/2}d_{5}\xi+B^{1/6}d_{6}\chi+B^{1/3}C^{1/2}d_{7}\tau+B^{1/3}C^{1/2}d_{8}\right)^{2}\\\
&+B^{2/3}Cd_{9}\Bigg{)}\end{split}$ (5.14)

where

$A=\frac{-2b_{1}V^{2}+3a_{1}}{2a_{1}V},B=\frac{V}{2a_{1}\beta},C=\frac{-V}{2},$
$a_{1}=\frac{(1+q)}{2},b_{1}=\frac{(1+q)(3-q)}{8}$

## Particular Parameter Value

Set 1: $d_{1}=1,d_{2}=a,d_{5}=0,d_{6}=b,d_{4}=c,d_{8}=d$. Then

$g=\frac{\xi}{B^{1/3}}+\frac{a\chi}{B^{1/6}C^{1/2}}+(a^{2}-b^{2})\tau+c$
(5.15) $h=\frac{b\chi}{B^{1/6}C^{1/2}}+2ab\tau+d$ (5.16)
$\begin{split}f=&\left(\frac{\xi}{B^{1/3}}+\frac{a\chi}{B^{1/6}C^{1/2}}+(a^{2}-b^{2})\tau+c\right)^{2}\\\
&+\left(\frac{b\chi}{B^{1/6}C^{1/2}}+2ab\tau+d\right)^{2}+\frac{3}{b^{2}}\end{split}$
(5.17)

Hence the lump solutions

$\phi(\xi,\chi,\tau)=4(\frac{6B^{1/3}}{A})\frac{-\left(\frac{\xi}{B^{1/3}}+\frac{a\chi}{B^{1/6}C^{1/2}}+(a^{2}-b^{2})\tau+c\right)^{2}+\left(\frac{b\chi}{B^{1/6}C^{1/2}}+2ab\tau+d\right)^{2}+\frac{3}{b^{2}}}{\Bigg{(}\left(\frac{\xi}{B^{1/3}}+\frac{a\chi}{B^{1/6}C^{1/2}}+(a^{2}-b^{2})\tau+c\right)^{2}+\left(\frac{b\chi}{B^{1/6}C^{1/2}}+2ab\tau+d\right)^{2}+\frac{3}{b^{2}}\Bigg{)}^{2}}$
(5.18)

Under the consideration of $c=d=0$, (5.18) reduces to

$\phi(\xi,\chi,\tau)=4(\frac{6B^{1/3}}{A})\frac{-\left(\frac{\xi}{B^{1/3}}+\frac{a\chi}{B^{1/6}C^{1/2}}+(a^{2}-b^{2})\tau\right)^{2}+b^{2}\left(\frac{\chi}{B^{1/6}C^{1/2}}+2a\tau\right)^{2}+\frac{3}{b^{2}}}{\Bigg{(}\left(\frac{\xi}{B^{1/3}}+\frac{a\chi}{B^{1/6}C^{1/2}}+(a^{2}-b^{2})\tau\right)^{2}+b^{2}\left(\frac{\chi}{B^{1/6}C^{1/2}}+2a\tau\right)^{2}+\frac{3}{b^{2}}\Bigg{)}^{2}}$
(5.19)

Set 2: $d_{1}=1,d_{2}=2,d_{4}=0,d_{5}=1,d_{6}=-1,d_{8}=0$. Then

$g=\frac{\xi}{B^{1/3}}+\frac{2\chi}{B^{1/6}C^{1/2}}-\frac{1}{2}\tau$ (5.20)
$h=\frac{\xi}{B^{1/3}}-\frac{\chi}{B^{1/6}C^{1/2}}-\frac{7}{2}\tau$ (5.21)
$f=\frac{25}{2}\tau^{2}-8\frac{\xi\tau}{B^{1/3}}+5\frac{\chi\tau}{B^{1/6}C^{1/2}}+2\frac{\xi^{2}}{B^{2/3}}+5\frac{\chi^{2}}{B^{1/3}C}+2\frac{\xi\chi}{B^{1/2}C^{1/2}}+\frac{8}{3}$
(5.22)

So, the lump solutions

$\phi(\xi,\chi,\tau)=48(\frac{6B^{1/3}}{A})\frac{\left(-21\tau^{2}+48\frac{\xi\tau}{B^{1/3}}+78\frac{\chi\tau}{B^{1/6}C^{1/2}}-12\frac{\xi^{2}}{B^{2/3}}+24\frac{\chi^{2}}{B^{1/3}C}-12\frac{\xi\chi}{B^{1/2}C^{1/2}}+16\right)}{\left(75\tau^{2}-48\frac{\xi\tau}{B^{1/3}}+30\frac{\chi\tau}{B^{1/6}C^{1/2}}+12\frac{\xi^{2}}{B^{2/3}}+30\frac{\chi^{2}}{B^{1/3}C}+12\frac{\xi\chi}{B^{1/2}C^{1/2}}+16\right)^{2}}$
(5.23)

Set 3: $d_{1}=1,d_{2}=-2,d_{4}=0,d_{5}=-2,d_{6}=1,d_{8}=0.$ Then

$g=\frac{\xi}{B^{1/3}}-\frac{2\chi}{B^{1/6}C^{1/2}}+\frac{11}{5}\tau$ (5.24)
$h=\frac{-2\xi}{B^{1/3}}+\frac{\chi}{B^{1/6}C^{1/2}}+\frac{2}{5}\tau$ (5.25)
$f=\frac{5\xi^{2}}{B^{2/3}}+\frac{5\chi^{2}}{B^{1/3}C}+5\tau^{2}-\frac{8\xi\chi}{B^{1/2}C^{1/2}}-\frac{8\chi\tau}{B^{1/6}C^{1/2}}+\frac{14}{5}\frac{\xi\tau}{B^{1/3}}+\frac{125}{3}$
(5.26)

Hence the lump solutions are given as follows

$\phi(\xi,\chi,\tau)=12(\frac{6B^{1/3}}{A})\frac{1581\tau^{2}-\frac{1875\xi^{2}}{B^{2/3}}-\frac{525\chi^{2}}{B^{1/3}C}+\frac{3000\xi\chi}{B^{1/2}C^{1/2}}-\frac{1320\chi\tau}{B^{1/6}C^{1/2}}-\frac{1050\xi\tau}{B^{1/3}}+15625}{\left(\frac{75\xi^{2}}{B^{2/3}}+\frac{75\chi^{2}}{B^{1/3}C}+75\tau^{2}-\frac{120\xi\chi}{B^{1/2}C^{1/2}}-\frac{120\chi\tau}{B^{1/6}C^{1/2}}+\frac{42\xi\tau}{B^{1/3}}+625\right)^{2}}$
(5.27)

##  6 Results and Discussions

In this work, a renowned nonlinear evolution equation, known as KP equation,
has been derived by applying Reductive Perturbation Technique (RPT) from a
dust hydrodynamical model. Then using the HBM, lump soliton solutions have
been derived. Here three different set of parameter values have been chosen
and accordingly, three lump soliton solutions arrived. Next we plot these
solutions and discuss the effects of plasma parameters on these.

Figure 1: Lump soliton solutions for the parameters, $a=1$, $b=1$,
$\beta=0.2$, $\delta_{1}=1.005$, $q=1.05$, $q=1.45$, $q=1.85$ at $\tau=0$

Figure 2: Contour plot of lump solutions, for $a=1$, $b=1$,$\beta=0.2$,
$\delta_{1}=1.005$, $q=1.05$, $q=1.45$, $q=1.85$ at $\tau=0$

Figure 3: 3(a) and 3(b) are the 2-D plot of lump soliton solutions for $\phi$
vs $\xi$, and $\phi$ vs $\chi$ respectively, for $\beta=0.2$,
$\delta_{1}=1.005$, $q=1.55$ at $\tau=0$

For parameter set-1, lump solitons have been plotted for distinct values of
nonextensive parameter q at time $\tau=0$, remaining the other parameters as
fixed. Figures 1(a), 1(b), and 1(c) are the 3-D lump structures, and Figure
2(a), 2(b), and 2(c) describe the solution features in the phase diagram for
same parameter set. Figure (1) narrates that the amplitude of lump solitons
are gradually decreasing for increasing value of nonextensive parameter $q$.
Figure (2) describes the region where the system is close as well as
conservative for same parameter regime. Figures 3(a) and 3(b) are plotted to
explain the behavioural change of lump solitons more conveniently, for
$q=1.55$.Figure 3(a) is framed by $\phi$ and $\xi$ co-ordinates, for
$\chi=0.4$ (green line), $\chi=1$ (red line), $\chi=1.4$ (blue line).
Similarly, Figure 3(b) is plotted for $\phi$ with respect to the coordinate
axis $\chi$ for $\xi=0.4$ (green line), $\xi=1$ (red line), $\xi=1.4$ (blue
line).

Figure 4: Lump soliton solutions, for the parameters, $\beta=0.2$,
$\delta_{1}=1.005$, $q=1.05$, $q=1.45$, $q=1.85$ at $\tau=0$

Figure 5: Contour plot of lump solitons, for $\beta=0.2$, $\delta_{1}=1.005$,
$q=1.05$, $q=1.45$, $q=1.85$ at $\tau=0$

Figure 6: Lump soliton solutions, for the parameters, $\beta=0.2$,
$\delta_{1}=1.005$, $q=1.85$ at $\tau=0$, $\tau=1$, $\tau=2$

Figure 7: 7(a) and 7(b) are the 2-D profile of lump solitons, for $\phi$ vs
$\xi$ and $\phi$ vs $\chi$ respectively, for $\beta=0.2$, $\delta_{1}=1.005$,
$q=1.55$ at $\tau=0$

Figures (4)-(7) have plotted for parameter set-2, to see the features of lump
solitons for distinct values of nonextensive parameter q, keeping rest of the
parameter as constant at $\tau=0$. Figure (4) describes the lump solitons and
Figure (5) shows the solution features in the phase diagram for $q=1.05$,
$q=1.45$, $q=1.85$ respectively. From Figure (4), it has been observed that,
the amplitude of lump solitons are gradually increasing for increasing values
of nonextensive parameter $q$. Figure (5) displays the region where the system
is close as well as conservative for same parameter regime. Figure (6) is the
lump structures at various time $\tau=0$, $\tau=1$, and $\tau=2$ respectively.
Figures 7(a), and 7(b) are the 2-D profile of lump structures, describe
strongly the behaviourial change of solitons for $q=1.55$. In Figure 7(a), the
electrostatic potential ($\phi$) has been drawn with respect to the co-
ordinate axis $\xi$, for $\chi=2$ (green line), $\chi=1$ (red line),
$\chi=1.4$ (blue line). Similarly, we have plotted $\phi$ vs $\chi$ graph in
7(b), for different frame i.e. for $\xi=1.6$ (green line), $\xi=1$ (red line),
$\xi=1.4$ (blue line).

Figure 8: Lump soliton solutions, for the parameters, $\beta=0.2$,
$\delta_{1}=1.005$, $q=1.05$, $q=1.45$, at $\tau=0$

Figure 9: Contour plot of lump solitons, for $\beta=0.2$, $\delta_{1}=1.005$,
$q=1.05$, $q=1.45$, at $\tau=0$

Figure 10: 10(a) and 10(b) are the 2-D profile of lump solitons, for $\phi$ vs
$\xi$ and $\phi$ vs $\chi$ respectively, for $\beta=0.2$, $\delta_{1}=1.005$,
$q=1.55$ at $\tau=0$

For parameter set-3, we have plotted Figures (8)-(10), to see the lump
solitons for various non-extensive
parameter q at time $\tau=0$ and keeping rest of the parameters as constant.
Figure 8(a), and 8(b) are the lump soliton structures and Figure 9(a), and
9(b) are the phase diagram for $q=1.05$, and $q=1.45$ respectively. It is
clear from Figures 8(a), and 8(b) that, when the nonextensive parameter
increases, the amplitude of lump solitons decreases gradually. Here Figures
9(a), and 9(b) show the region where the system is close as well as
conservative for same parameter set. To study the soliton structures in more
convenient way, we have plotted the soliton structures in one space dimension.
To characterize the change of soliton structures effectively, Figure (10) has
been plotted, for $q=1.55$ in one space dimension only. Figure 10(a) is the
$\phi$ vs $\xi$ plot, for $\chi=2$ (green line), $\chi=1$ (red line),
$\chi=1.4$ (blue line) and Figure 10(b) is the $\phi$ vs $\chi$ graph, for
$\xi=1.6$ (green line), $\xi=1$ (red line), $\xi=1.4$ (blue line).

DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

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