# Impact of the purposeful kinesis on running waves

N. Çabukoǧlu<EMAIL_ADDRESS>Department of Mathematics, University of
Leicester, Leicester, LE1 7RH, UK

###### Abstract

The basic model of purposeful kinesis was developed recently (Gorban &
Çabukoǧlu, Ecol. Complex. 2018, 33, 75–83) on the basis of the “Let well
enough alone” idea. According to this model the diffusion drops while the
reproduction coefficient is increasing. That is, species prefer to stay in a
good condition and the population gives birth; otherwise, in the bad situation
individuals want to run away because of the fatal conditions. In this study,
we analyse the impact of the purposeful kinesis model on running waves. The
running waves in the population with kinesis are studied using numerical
experiments. Both monotonic and non-monotonic (Allee effect) dependence of the
reproduction coefficient on the population density are studied. The possible
benefits of the purposeful kinesis are demonstrated: with the higher
diffusion, while the population without kinesis ends up with extinction, the
population with kinesis stays alive and has the running wave behaviour. While
the kinesis of the prey population is decreasing, the wave amplitude gets
smaller. On the other hand, for the lower kinesis of predators they have a
sharp increase.

###### keywords:

kinesis , diffusion , population , extinction , Allee effect , travelling
waves

##  1 Introduction

In this study, we have analysed the diffusion model with kinesis. Kinesis is
the non-directional movement according to the change of the local environment.
That is, the organism gets the information from the living area, and therefore
prefer to stay or move to the other beneficial place. Taxis is also related to
mobility. However, taxis requires the non-local information. The organism with
taxis has the directed movement towards to the condition or opposite side of
the stimulus.

We aim to explore the impact of purposeful kinesis on running waves using PDE
models. The classical PDE model of population dispersal was proposed by Patlak
[1953] and Keller and Segel [1971]. It was used to model taxis behaviour
[Hillen and Painter, 2009]. The reaction-diffusion model can be used for
kinesis modelling. The diffusion coefficient can depend on the local
situation. Cosner [2014] studied the reaction-diffusion models on animal
dispersal. He presented the study on animal movement: which is the better
condition to stay or leave, being slow or random movement, increasing the
population size is harmful or not.

In this study, we employ the purposeful kinesis model [Gorban and Cabukoglu,
2018a]. Often ‘Purposeful’ means intentionality that individuals are unable.
Rosenblueth and Wiener [1950] developed the general concept of purposeful
behaviour. ‘Purpose’ brings the optimization idea and this concept requires
the evolutionary optimality [Parker and Smith, 1990].

There is a connection between the reproduction rate and diffusion coefficient.
Average reproduction coefficient was defined as Darwinian fitness [Haldane,
1932, Metz et al., 1992, Gorban, 2007]. Migration should increase Darwinian
fitness.

In this paper, we analysed impact of the purposeful kinesis on running waves.
In particular, it was presented that the population with Allee effect can
spread late but with a higher diffusion the population will stay alive, and in
time will continue the running wave behaviour. We did numerical experiments
that kinesis model holds spreading invaded area, while the population without
kinesis dies.

##  2 Main Results

In this section, we will give the main results.

###  2.1 Running Waves

The kinesis model was selected in the following form [Gorban and Cabukoglu,
2018a]:

$\boxed{\partial_{t}u_{i}(x,t)=D_{0i}\nabla\cdot\left(e^{-\alpha_{i}r_{i}(u_{1},\ldots,u_{k},s)}\nabla
u_{i}\right)+r_{i}(u_{1},\ldots,u_{k},s)u_{i},}$ (1)

where:

  * $u_{i}$ is the $i$th species-population density,

  * $s$ is the abiotic characteristics of the living conditions,

  * $r_{i}$ is the reproduction coefficient,

  * $D_{0i}>0$ is the equilibrium diffusion coefficient which is defined when the reproduction coefficient is 0,

  * $\alpha_{i}>0$ defines the relation between the diffusion coefficient on the reproduction coefficient.

$D_{i}=D_{0i}e^{-\alpha r_{i}}$ characterizes the diffusion depending on
reproduction coefficient. According to that model, the diffusion depends on
well-being which is measured by reproduction coefficient.

Gorban and Cabukoglu [2018b] later studied the cost of mobility. The changes
of reproduction coefficient give the mobility cost value. If we take into
account the cost of mobility then the bifurcation is predicted: when the
conditions are getting worse, the mobility increases to a threshold level, and
then it vanishes. Therefore, in the worse conditions, there is no solution for
mobility. This mobility cost equation can be solved with Lambert $W$-function.

We will present below how the dependence of diffusion on well-being effects
the running waves of the population density on space.

Let us compare two models:

  * 1.

The PDE model for population with the constant diffusion coefficient: without
kinesis, KPP (Kolmogorov, Petrovsky and Piskunov, 1937):

$\partial_{t}u(t,x)=D\nabla^{2}u+(1-u(t,x))u(t,x),\\\ $ (2)

  * 2.

The PDE model for population with kinesis (KPP with modified diffusion
coefficient [KPP, 1937]):

$\partial_{t}u(t,x)=D\nabla\cdot\left(e^{-\alpha(1-u(t,x))}\nabla
u\right)+(1-u(t,x))u(t,x),\\\ $ (3)

We have first used MATLAB pdepe [2017] function to solve one dimensional
system of PDE. Then, we used the MATHEMATICA NDSolve [2014] solver which is
applied the Hermite method and Newton’s divided difference formula to solve
two dimensional system.

We selected the space on the interval $[-50,50]$ with Dirichlet boundary
conditions and with the initial conditions:

$u(0,x)=\dfrac{1}{1+e^{\lambda x}}.$

The values of the constants are: $D=1$, $\alpha=1$, $\lambda=10$.

Figure 1: Travelling waves of Population density: a) for animals without
kinesis for model (2) and b) for animals with kinesis for model (3). The
values of the constants are: $D=1$, $\alpha=1$, $\lambda=10$.

In the large time the running waves converge to a unique front with the
velocity $v$. For example, the logistic growth model $r(u)=\alpha
u\left(1-u/K\right)$ speed will be $v_{0}=2\sqrt{\alpha D}$.

Fig. 1 presents that the proposed ‘minimal purposeful kinesis model’ has the
running wave behaviour on space. In these given conditions, the waves in the
population model with kinesis is slower which is expalined in Fig. 2. When we
increase the kinesis parameter $\alpha$, the spreading velocity will decrease
monotonically.

Figure 2: Wave velocity as function of the kinesis parameter $\alpha$ for
model (3).

We have done numerical experiments on MATLAB to see the relations between the
wave velocity and the kinesis parameter $\alpha$ (see Fig. 2). While $\alpha$
is increasing, the velocity starts to decrease monotonically. After
$\alpha=6$, the velocity is almost stabilizing. That is, the population
spreading on space can be almost the same in the different time lines.

###  2.2 Running Waves with the Allee effect

Allee effect was introduced as nonmonotonic behaviour of the reproduction
coefficient as a function of population density. At the beginning, the
reproduction rate is minimum and then reaches to the peak. After the maximum
reproduction rate, it starts to decrease monotonically while the population
size is increasing [Allee et al., 1949]. There are a lot of reasons why
populations show Allee effect such as low rate feeding [Way and Banks, 1967,
Way and Cammell, 1970], and for some reasons the population reduces the
predator defence [Kenward, 1978, Kruuk, 1964], give up the breeding for
psychological reasons [Ralls et al., 1986, Soule et al., 1986], and a wide
variety of reasons [Caraco and Pulliam, 1984, Folt, 1987, Foster and Treherne,
1981, Turchin and Kareiva, 1989].

Lewis and Kareiva [1993] explored the population spread of asymptotic rates
with the Allee effect. In two dimensional analysis, they investigated that the
patterns of spread have been influenced by this population model. We have used
the coefficients in that study to analyse how the population with kinesis
affects the two-dimensional patterns of spread.

The basic form of the reproduction coefficient with Allee effect is

$r(u)=r_{0}(K-u)(u-\beta).$

The models with purposeful kinesis on Allee effect were studied by Gorban and
Cabukoglu [2018a] previously.

Gorban and Sadovskiy [1989] introduced the study that when the average
population size is less than the optimum density, the evolutionary optimal
strategy for populations with the Allee effect collapses with optimal density.
There are some other consequences of Allee effect with diffusion: the
population spreads in a massive way through the formation, the interaction
between species and the movement of separate patches even if the population is
inhomogeneous external condition [Morozov et al., 2006, Petrovskii et al.,
2002].

We may present the reaction-diffusion equations for a single population with
the Allee effect for a system without kinesis (4) and for a system with
kinesis (5).

$\partial_{t}u(t,x)=D\nabla\cdot(\nabla u)+k(1-u)(u-\beta)u(t,x),\\\ $ (4)
$\partial_{t}u(t,x)=D\nabla\cdot\left(e^{-\alpha k(1-u)(u-\beta)}\nabla
u\right)+k(1-u)(u-\beta)u(t,x).\\\ $ (5)

The values of constants are: $D=1$, $\alpha=1$, $\beta=0.7$.

Eqs. (4) and (5) are solved for one space variable $x\in[-50,50]$ with
Dirichlet boundary conditions and with the initial conditions:

$u(0,x)=\dfrac{1}{1+e^{\lambda x}};\lambda=10.$ (6)

We choose $k=k(\beta)$ as the normalization constant which was given by Lewis
and Kareiva [1993] as determined by the maximum growth rate. $k$ has been
determined as

$k=1;$ (7) $k=4/(1-\beta)^{2};$ (8)
$k=27/\left(2\left(\left(\left(1+\beta\right)^{2}-9\beta/2\right)\left(1+\beta\right)+\left(\left(1+\beta\right)^{2}-3\beta\right)^{3/2}\right)\right).$
(9)

It is considered that travelling wave solutions to Eqs. (4) and (5) are in the
form of $u=U(z)$ with $z=x-vt$, where $v$ is the velocity.

It was shown that there is a unique solution to these Eqs. (4) and (5) with a
unique $v$ velocity with the condition $0<\beta<1$ by Fife [1979]. When the
Allee effect parameter is negative, there is a minimum value of velocity such
that there exists a corresponding running wave solution [Aronson and Weinberg,
1978, Fife, 1979, Hadeler and Rothe, 1975]. After a long time period
$t\rightarrow\infty$, the solution of the system 4 may converge to a
travelling wave solution.

Figure 3: The running waves of population densities with the Allee effect: (a)
for animals without kinesis (4), (b) for animals with kinesis in time 0 to 100
(5). The values of constants are: $D=1$, $\alpha=1$, $\beta=0.7$.

The numerical results (Fig. 3) show that the population with Allee effect has
the running wave behaviour both with kinesis and without kinesis. Population
with kinesis demonstrates faster running waves on space. Fast-spreading effect
of the population could also end up to extinction for the population with
Allee effect. When the Allee effect is higher than the population density, the
reproduction coefficient becomes negative. Therefore, this leads the
population to extinction.

a) b) c)

Figure 4:  Allee effect parameter and speed $\alpha=1$ and $D=1$. The
population with kinesis and without kinesis a) the normalization constant is
$k=1$ for models (4) and (4), b) $k$ is given for model (8). c) $k$ is given
for model (9). The system has been solved with the numerical method for the
running wave front $u=0.5$

Wave velocity for different models has been displayed as the Fig. 4 with the
Allee effect parameter $-1\leq\beta\leq 0.5$. The Allee effect decreases the
rate of spread of an invading population. We can see from the Fig. 4 that with
these conditions the spreading of the population with the Allee effect is
decreasing when $\beta$ is getting higher value. Moreover, the spreading
velocity of the population without kinesis is faster than the population with
kinesis.

###  2.3 Two-Dimensional Spread

Analytical representation of two-dimensional system may be too complex to
solve. Aronson and Weinberg [1978] presented that the planar travelling waves
exist and this form can be used by an invading population. They studied the
two-dimensional population model with Allee dynamics to see the invaded area
of the population. The population, with Allee effect, may end up with
extinction. This can happen even if the population density exceeds the
threshold. The reason is that reproduction, the population growth, cannot be
sufficient to prevent the population dispersal. There are some critical
factors to determine the invading success: shape and size of the environment.
It has been already shown that how the population stay alive between invaded
areas and unoccupied places [Lewis and Kareiva, 1993].

Now, in this section, we present the two-dimensional model that we want to see
the invaded and unoccupied areas of the population with kinesis and without
kinesis. In addition to that model, we discuss the population with Allee
effect.

a) b)

Figure 5: Initial distribution ($t=0$):
$u(0,x)=\frac{1}{1+e^{\frac{x^{2}+y^{2}}{2}}}.$ The invaded area by the
population with kinesis and without kinesis.

Fig. 5 shows the initial distribution that the invaded area is the same for
the population with kinesis and without kinesis.

a) b) c)

Figure 6:  2D Population without kinesis $\alpha=1$ and $D=1$.

a) b) c)

Figure 7:  2D Population with kinesis $\alpha=1$ and $D=1$.

a) b) c)

Figure 8:  2D Allee effect without kinesis $\alpha=1$, $D=1$, $\beta=0.05$.

a) b) c)

Figure 9:  2D Allee effect with kinesis $\alpha=1$, $D=1$, $\beta=0.05$.

a) b) c)

Figure 10:  Invaded areas at center by the population without kinesis and
Allee effect. The parameters are: $\alpha=1$, $D=1$, $\beta=0.05$.

a) b)

Figure 11:  Invaded areas at center by the population with kinesis and Allee
effect. At time $t=16$, the population dies out and there is no waves. The
parameters are: $\alpha=1$, $D=1$, $\beta=0.05$.

(a) $t=4$ (b) $t=8$ (c) $t=16$

Figure 12:  2D Allee effect without kinesis $\alpha=5$, $D=0.5$, $\beta=0.05$
with Neumann boundary conditions.

(a) $t=4$ (b) $t=8$ (c) $t=16$

Figure 13:  2D Allee effect with kinesis $\alpha=5$, $D=0.5$, $\beta=0.05$
with Neumann boundary conditions.

(a) $t=4$ (b) $t=8$

Figure 14:  Invaded areas at center by the population without kinesis and
Allee effect. At time $t=16$, there will be no waves since the population
dies. The parameters are: $\alpha=5$, $D=0.5$, $\beta=0.05$.

(a) $t=4$ (b) $t=8$ (c) $t=16$

Figure 15:  Invaded areas at center by the population with kinesis and Allee
effect. The parameters are: $\alpha=5$, $D=0.5$, $\beta=0.05$.

##  3 Predator-Prey Running Waves

In this section, we consider competing species in the same environment. These
two species interact with the model without kinesis and the new model with
kinesis. Volpert and Petrovskii analysed the running waves of predator-prey
interaction model in their study [Volpert and Petrovskii, 2009]. Now, we
compare the population with kinesis and without kinesis as running wave
behaviour.

  * 1.

The PDE model for population with the constant diffusion coefficient (i.e.
without kinesis):

$\displaystyle\partial_{t}u(t,x)$ $\displaystyle=$ $\displaystyle
D\nabla\cdot(\nabla u)+r_{u}u(t,x),$ (10) $\displaystyle\partial_{t}v(t,x)$
$\displaystyle=$ $\displaystyle D\nabla\cdot(\nabla u)+r_{v}v(t,x),$ (11)

  * 2.

The PDE model for population with kinesis:

$\displaystyle\partial_{t}u(t,x)$ $\displaystyle=$ $\displaystyle
D\nabla\cdot\left(e^{-\sigma_{u}r_{u}}\nabla u\right)+r_{u}u(t,x),$ (12)
$\displaystyle\partial_{t}v(t,x)$ $\displaystyle=$ $\displaystyle
D\nabla\cdot\left(e^{-\sigma_{v}r_{v}}\nabla u\right)+r_{v}v(t,x).$ (13)

We consider the simplest predator-prey model in the classical Lotka-Volterra
form in dimensionless variables $r_{u}=k-v$ and $r_{v}=u-m$; $u$ and $v$ are
the population densities of prey and predator.

Eqs. (10), (11), (12) and (13) are solved for one space variable
$x\in[-100,100]$ with Dirichlet boundary conditions and with the initial
conditions:

$u(0,x)=v(0,x)=\dfrac{1}{1+e^{\lambda x}};\lambda=10.$ (14)

a) b)

Figure 16: Travelling waves of predator-prey diffusion (a) for animals without
kinesis,(10, 11) $\sigma_{1}=\sigma_{2}=1$ (b) for animals with kinesis (12,
13) $\sigma_{1}=\sigma_{2}=1$. The values of constants are: $D=1$, $k=1$,
$m=1$.

a) b)

Figure 17: Travelling waves of predator-prey diffusion (a) for predators with
kinesis and preys without kinesis $\sigma_{1}=0,$ $\sigma_{2}=1.$ (b) for
predators without kinesis and preys with kinesis $\sigma_{1}=1,$
$\sigma_{2}=0.$ The values of constants are: $D=1$, $k=1$, $m=1$.

The numerical simulations with the 1D case show that depending on the values
the systems (10), (11), (12) and (13) show a variety of travelling fronts.

We compare four cases with Figs. 16 and 17:

  * 1.

both predator and prey without kinesis,

  * 2.

both predator and prey with kinesis,

  * 3.

predator without kinesis and prey with kinesis,

  * 4.

predator with kinesis and prey without kinesis.

All of them have the running wave behaviour. We demonstrate with Fig. 16b that
the predator and prey populations with kinesis have the running wave behaviour
and the peaks between predator and prey increase. For the predator without
kinesis and prey with kinesis (Fig. 17a), the amplitude decreases. On the
other hand, the predator with kinesis and prey without kinesis (Fig. 17b)
running wave peaks of the population density become sharper.

##  4 Discussion

It has been suggested a purposeful kinesis model that the diffusion
coefficient depends on well-being which is measured by the reproduction
coefficient by Gorban and Cabukoglu [2018a]. The system formalizes the simple
rule: “Let well enough alone”. Volpert and Petrovskii presented the study on
travelling waves of the reaction-diffusion model Volpert and Petrovskii
[2009].

In this paper, we analysed the running wave behaviour of the reaction-
diffusion model with kinesis. The running wave theory begins with Fisher
[1930] and Kolmogorov, Petrovskii and Piskunov [KPP, 1937]. They defined the
travelling wave solutions of the scalar reaction-diffusion equation and
analysed their existence, stability and the speed of these waves. Lewis and
Kareiva explored the relevance between the Allee effect parameter $\beta$ and
the relative wave speed. They also studied on the two-dimensional spread of
invading populations with Allee effect.

In this study, we have demonstrated the speed of the running waves of the
population with kinesis. These following have been observed:

  * 1.

While the relation between the diffusion coefficient and reproduction
coefficient $\alpha$ is increasing the running wave speed monotonically
decreases, and at some point stabilizes (see Fig. 2).

  * 2.

The Allee effect parameter $\beta$ and velocity $v$ have been displayed by
Fig. 4 for the values of normalization coefficient $k$ on the running wave
front $u=0.5$. While the Allee effect parameter is increasing, the
populations’ running speed decreases. When $\beta=0.5$, the velocity is 0.
After that value of $\beta$, the waves of the population model show the
running wave behaviour through the left hand side. The individuals with
kinesis run slower than the population without kinesis. This may lead up an
extinction in the invaded area.

  * 3.

The initial distributions of the invaded area are the same for both
populations (see Fig. 5). System stabilization is observed by $r=0$.
Therefore, Kinesis does not change the stability of homogeneous positive
steady-states.

  * 4.

The invasion of the population with kinesis occupies the larger area than
without kinesis. That is, they stay alive in the habitat.

  * 5.

Allee dynamics can affect the population extinction (see Fig. 14). On the
other hand, with the higher diffusion, the population with kinesis do not
extinct and can invade a large area (see Fig. 15).

  * 6.

We have also presented the running wave behaviour of the predator-prey model
with kinesis. The constant diffusion model waves were recently analysed by
Volpert and Petrovskii [2009] for the predator-prey model. We have observed
also that this model can have running wave behaviour. Kinesis of prey
decreases amplitude of the wave smaller, whereas kinesis of the predators can
significantly increase the peak value (Fig. 17b).

These population models with kinesis can be adapted to several models. It can
be seen that individuals may have the running wave behaviour in the invaded
areas.

## Acknowledgement

I am grateful to A.N. Gorban for his advises.

## References

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