# Basic Model of Purposeful Kinesis

A.N. Gorban<EMAIL_ADDRESS>N. Çabukoǧlu<EMAIL_ADDRESS>Department of
Mathematics, University of Leicester, Leicester, LE1 7RH, UK  Lobachevsky
University, Nizhni Novgorod, Russia

###### Abstract

The notions of taxis and kinesis are introduced and used to describe two types
of behaviour of an organism in non-uniform conditions: (i) Taxis means the
guided movement to more favorable conditions; (ii) Kinesis is the non-
directional change in space motion in response to the change of conditions.
Migration and dispersal of animals has evolved under control of natural
selection. In a simple formalisation, the strategy of dispersal should
increase Darwinian fitness. We introduce new models of purposeful kinesis with
diffusion coefficient dependent on fitness. The local and instant evaluation
of Darwinian fitness is used, the reproduction coefficient. New models include
one additional parameter, intensity of kinesis, and may be considered as the
minimal models of purposeful kinesis. The properties of models are explored by
a series of numerical experiments. It is demonstrated how kinesis could be
beneficial for assimilation of patches of food or of periodic fluctuations.
Kinesis based on local and instant estimations of fitness is not always
beneficial: for species with the Allee effect it can delay invasion and
spreading. It is proven that kinesis cannot modify stability of positive
homogeneous steady states.

###### keywords:

kinesis , diffusion , fitness , population , extinction , Allee effect

##  1 Introduction

The notions of taxis and kinesis are introduced and used to describe two types
of behaviour of an organism in non-uniform conditions:

  * 1.

Taxis means the guided movement to more favorable conditions.

  * 2.

Kinesis is the non-directional change in space motion in response to the
change of conditions.

In reality, we cannot expect pure taxis without any sign of kinesis. On the
other hand, kinesis can be considered as a reaction to the local change of
conditions without any global information about distant sites or concentration
gradients. If the information available to an organisms is completely local
then taxis is impossible and kinesis remains the only possibility of
purposeful change of spatial behaviour in answer to the change of conditions.
The interrelations between taxis and kinesis may be non-trivial: for example,
kinesis can facilitate exploration and help to find non-local information
about the living conditions. With this non-local information taxis is
possible.

In this paper, we aim to present and explore a simple but basic model of
purposeful kinesis. Kinesis is a phenomenon observed in a wide variety of
organisms, down to the bacterial scale. Purposeful seems to imply a sort of
intentionality that these organisms are incapable of. The terms ‘purpose’ and
‘purposeful’ are used in mathematical modelling of biological phenomena in a
wider sense than in psychology. ‘Purpose’ appears in a model when it includes
optimisation. The general concept of purposeful behaviour [Rosenblueth and
Wiener, 1950] of animals requires the idea of evolutionary optimality [Parker
and Smith, 1990]. In many cases this optimality can be deduced from kinetic
equations in a form of maximization of the average in time reproduction
coefficient – Darwinian fitness [Metz et al., 1992, Gorban, 1984, 2007].
Application of this idea to optimization of behaviour is the essence of
evolutionary game theory and its applications to population dynamics [Hofbauer
and Sigmund, 1998].

There are three crucial questions for creation of an evolutionary game model:

  1. 1.

Which information is available and usable? Dall et al. [2005] proposed a
quantitative theoretical framework in evolutionary ecology for analysing the
use of information by animal. Nevertheless, the question about information
which can be recognised, collected and used by an animal requires empirical
answers. Answering this question may be very complicated for analysis of
taxis, which involves various forms of non-local information. For kinesis the
situation is much simpler: the point-wise values of several fields
(concentrations or densities) are assumed to be known [Sadovskiy et al.,
2009].

  2. 2.

What is the set of the available behaviour strategies? All the organisms, from
bacteria to humans have their own set of available behaviour strategies, and
no organism can be omnipotent. It is necessary to describe constructively the
repertoire of potentially possible behaviours.

  3. 3.

What are the statistical characteristics of the environment, in particular,
and what are the laws and correlations in the changing of environment in space
and time? It is worth mentioning that all the changes in the environment
should be measured by the corresponding changes of the reproduction
coefficient.

We use a toy model to illustrate the idea of purposeful kinesis. Assume that
an animal can use one of two locations for reproduction. Let the environment
in these locations can be in one of two states during the reproduction period,
$A$ or $B$. The number of survived descendants is $r_{A}$ in state $A$ and
$r_{B}$ in state $B$. After that, the offsprings leave the area of
reproduction and their further survival does not depend on this area. Assume
also that the change of states can be described by a Markov chain with
transition probabilities $P_{A\to B}=p$ and $P_{B\to A}=q$. These assumptions
answer Question 3.

The animal is assumed to be very simple: it can just evaluate the previous
state of the location where it is now but cannot predict the future state.
There is no memory: it does not remember the properties of the locations where
it was before. This is the answer to question 1.

Finally, there is only one available behaviour strategy: to select the current
(somehow chosen) location or to move to another one. There exists resources
for one jump only and no ‘oscillating’ jumps between locations are possible.
This means that after the change of location the animal selects the new
location for reproduction independently of its state. Thus Question 2 is
answered.

Analysis of the model is also simple. If the state of the location is unknown
then the probability of finding it in state $A$ is $\frac{q}{p+q}$ and the
probability of finding it in state $B$ is $\frac{p}{p+q}$; these are the
stationary probabilities of the Markov chain. The expectation of the number of
offsprings without arbitrary information is

$r_{0}=\frac{qr_{A}+pr_{B}}{p+q}.$

If an animal chooses for reproduction the location with the previous state $A$
then the conditional expectation of the number of offsprings is
$r\left|{}_{A}\right.=(1-p)r_{A}+pr_{B}$. If it chooses the location with the
previous state $B$ then the expected number of offsprings is
$r\left|{}_{B}\right.=(1-q)r_{B}+qr_{A}$.

If the animal is situated in the location with the previous state $A$, and
$r\left|{}_{A}\right.<r_{0}$, then the change of location will increase the
expected number of offsprings. Analogously, if it is situated in the location
with the previous state $B$, and $r\left|{}_{B}\right.<r_{0}$, then the change
of location will increase the number of offsprings.

We have obtained the simplest model with mobility dependent conditionally
expected reproduction coefficient $r\left|{}_{\bullet}\right.$ under given
local conditions: if $r\left|{}_{\bullet}\right.$ is less than the value
$r_{0}$ expected for the indefinite situation then jump, else stay in the same
location. This is the essence of purposeful kinesis for this toy model.

It is very difficult to find realistic space and time correlations in the
environment during the evolution of animals under consideration. The answers
to Questions 1 and 2 for real animals are also non-obvious, but the main idea
can be utilised for the modelling of kinesis. We expect that the dynamics of
the models could provide insight, regardless of whether parameters were
obtained from optimization of real Darwininan fitness or just the structure of
equations was guessed on the basis of this optimization.

In this paper, we study PDE models of space distribution. We start from the
classical family of models. Patlak [1953], and Keller and Segel [1971]
proposed a PDE system which is widely used for taxis modelling [Hillen and
Painter, 2009].

$\begin{split}&\partial_{t}u(t,x)=\nabla\left(k_{1}(u,s)\nabla
u+k_{2}(u,s)u\nabla s\right)+k_{3}(u,s)u,\\\
&\partial_{t}s(t,x)=D_{s}\nabla^{2}s+k_{4}(u,s)-k_{5}(u,s)s,\end{split}$ (1)

where

  * $u\geq 0$ is the population density,

  * $s\geq 0$ is the concentration of the attractant,

  * $D_{s}\geq 0$ is the diffusion coefficient of the attractant,

  * coefficients $k_{i}(u,s)\geq 0$.

Coefficient $k_{1}(u,s)$ is a diffusion coefficient of the animals. It depends
on the population density $u$ and on the concentration of the attractant $s$.
Coefficient $k_{2}(u,s)$ describes intensity of population drift.

Special random processes were introduced for ‘microscopic’ theory of dispersal
in biological systems by Othmer et al. [1988]. They consist of two modes: (i)
position jump or kangaroo processes, and (ii) velocity jump processes:

  * 1.

The kangaroo process comprises a sequence of pauses and jumps. The
distributions of the waiting time, the direction and distance of a jump are
fixed;

  * 2.

The velocity jump process consists of a sequence of ‘runs’ separated by
reorientations, after which a new velocity is chosen.

Equations (1) can be produced from kinetic (transport) models of velocity–jump
random processes [Othmer and Hillen, 2000, 2002] in the limit of large number
of animals and small density gradients under an appropriate scaling of space
and time. The higher approximations are also available in the spirit of the
Chapman–Enskog expansion from physical kinetics [Chapman and Cowling, 1970].
Chalub et al. [2004] found sufficient conditions of absence of finite-time-
blow-ups in chemotaxis models. Turchin [1989] demonstrated that attraction
(and repulsion) between animals could modify the space dispersal of population
if this interaction is strong enough. Méndez et al. [2012] derived reaction-
dispersal-aggregation equations from Markovian reaction-random walks with
density-dependent transition probabilities. They have obtained a general
threshold value for dispersal stability and found the sufficient conditions
for the emergence of non-trivial spatial patterns. Grünbaum [1999] studied how
the advection-diffusion equation can be produced for organisms (“searchers”)
with different food searching strategies with various turning rate and turning
time distributions, which depend on the density of observed food distribution.

The family of models (1) is rich enough and the term $\nabla(k_{1}(u,s)\nabla
u)$ can be responsible for modelling of kinesis: it describes non-directional
motion in space with the diffusion coefficient $D=k_{1}(u,s)$. This
coefficient depends on the local situation represented by $u$ and $s$. In some
sense, the family of models (1) is even too rich: it includes five unknown
functions $k_{i}$ with the only requirement, the non-negativity.

Cosner [2014] reviewed PDE reaction–advection–diffusion models for the
ecological effects and evolution of dispersal, and mathematical methods for
analyzing those models. In particular, he discussed a series of optimality or
evolutionary questions which arose naturally: Is it better for the predators
to track the prey density, the prey’s resources, or some kind of combination?
Is it more effective for predators to slow down their random movement when
prey are present or to use directed movement up the gradient of prey density?
Should either predators or prey avoid crowding by their own species? Cosner
[2014] presented also examples when diffusion is harmful for the existence of
species: if the average in space of the reproduction coefficient is negative
for all distributions of species then for high diffusion there is no steady
state with positive total population even if there exist steady states with
positive total population for zero or small diffusion (for connected areas).
The possibility of organisms moving sub- or super-diffusively, e.g. Lévy
walks, fractional diffusion, etc. (see, for example the works by Chen et al.
[2010], Méndez et al. [2010]), can be combined with the idea of purposeful
mobility (see, for example the works by Chen et al. [2010], Méndez et al.
[2010]) but we limit analysis in this paper by the classical PDE.

In this work, we study the population dispersal without taxis, therefore, the
advection coefficients $k_{2}$ is set below to zero. Such dispersal strategy
seems to be quite limited comparing to the general kinesis+taxis dispersal
system. Nevertheless, Nolting et al. [2015] demonstrated on the jump models
that the purely kinesis (non-directional dispersal strategy) allows foragers
to identify efficiently intensive search zones without taxis and are more
robust to changes in resource distribution.

We also assume strong connection between the reproduction coefficient
$r=k_{3}$ and the diffusion coefficient. The reproduction coefficient
characterises both the competitive abilities of individuals, and their
fecundity. The Darwinian fitness is the average reproduction coefficient in a
series of generations [Haldane, 1932, Metz et al., 1992, Gorban, 2007].
Dynamics maximise the Darwinian fitness of survivors (this is formalisation of
natural selection). Unfortunately, evaluation of this quantity is non-local in
time and requires some knowledge of the future. Therefore, we use below the
local in time and space estimation of fitness and measure the well-being by
the instant and local value of the reproduction coefficient $r$. This is a
rather usual approach but it should be used with caution: in some cases, the
optimisation of the local criteria can worsen the long-time performance. We
describe one such situation below: use a locally optimised strategy of kinesis
may delay invasion and spreading of species with the Allee effect. On another
hand, we demonstrate how kinesis controlled by the local reproduction
coefficient may be beneficial for assimilation of patches of food or periodic
fluctuations.

##  2 Main Results

###  2.1 The “Let well enough alone” model

The kinesis strategy controlled by the locally and instantly evaluated well-
being can be described in simple words: Animals stay longer in good conditions
and leave quicker bad conditions. If the well-being is measured by the instant
and local reproduction coefficient then the minimal model of kinesis can be
written as follows:

$\boxed{\partial_{t}u_{i}(x,t)=D_{0i}\nabla\left(e^{-\alpha_{i}r_{i}(u_{1},\ldots,u_{k},s)}\nabla
u_{i}\right)+r_{i}(u_{1},\ldots,u_{k},s)u_{i},}$ (2)

where:

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

  * $s$ represents the abiotic characteristics of the living conditions (can be multidimensional),

  * $r_{i}$ is the reproduction coefficient, which depends on all $u_{i}$ and on $s$,

  * $D_{0i}>0$ is the equilibrium diffusion coefficient (defined for $r_{i}=0$),

  * The coefficient $\alpha_{i}>0$ characterises dependence of the diffusion coefficient on the reproduction coefficient.

Equations (2) describe dynamics of the population densities for arbitrary
dynamics of $s$. For the complete model the equations for environment $s$
should be added. The space distribution strategy is summarised in the
diffusion coefficient $D_{i}=D_{0i}e^{-\alpha_{i}r_{i}}$, which depends only
on the local in space and time value of the reproduction coefficient.
Diffusion depends on well-being measured by this coefficient. We can see that
the new models add one new parameter per species to the equations (instead of
function $k_{1}(u,s)$ in (1)). This is the kinesis constant $\alpha_{i}$. It
can be defined as

$\alpha_{i}=-\frac{1}{D_{0i}}\left.\frac{dD_{i}(r_{i})}{dr_{i}}\right|_{r_{i}=0}.$

In the first approximation, $D_{i}=D_{0i}(1-\alpha_{i}r_{i})$. The exponential
form in (2) guarantees positivity of the coefficient $D_{i}$ for all values of
$r_{i}$.

For good conditions ($r_{i}>0$) diffusion is slower than at equilibrium
($r_{i}=0$) and for worse conditions ($r_{i}<0$) it is faster. Equations (2)
just formalise a simple wisdom: do not change the location that is already
good enough (let well enough alone) and run away from bad location.

We analyse below how the dependence of diffusion on well-being effects patch
dynamics and waves in population dynamics.

###  2.2 Stability of uniform distribution

The positive uniform steady state $(u^{*},s^{*})$ satisfies the equation:
$r_{i}(u^{*}_{1},\ldots,u^{*}_{k},s^{*})=0$.

The linearised equations near the positive uniform steady state are

$\partial_{t}\delta u_{i}(x,t)=d_{i}\nabla^{2}(\delta
u_{i})+u_{i}\left(\sum_{j}r_{i,j}\delta u_{j}+r_{i,0}\delta s\right),$ (3)

where

  * $\delta u_{i}$ is the deviation of the population density of $i$th animal from equilibrium $u^{*}$, $\delta s=s-s^{*}$,

  * $r_{i,j}=\partial r_{i}/\partial u_{j}|_{(u^{*},s^{*})}$, $r_{i,0}=\partial r_{i}/\partial s|_{(u^{*},s^{*})}$ are derivatives of $r$ at equilibrium.

These linearised equation are the same as for the system without kinesis (with
constant diffusion coefficients). Therefore, kinesis does not change stability
of positive uniform steady states. Moreover, near such a steady state
linearised equations for a system with kinesis are the same as for the system
with constant diffusion coefficient.

There is an important difference between possible dynamic consequences of
taxis and kinesis: we proved that kinesis cannot modify stability of
homogeneous steady states, whereas Tyutyunov et al. [2017] demonstrated that
taxis can destabilise them.

###  2.3 Utilisation of a patch of food

Figure 1: A schematic representation of a patch of food.

As a first test for the new model we used utilisation of a patch of food (a
sketch of this gedankenexperiment is presented in Fig. 1). Concentration of
food in patches is one of the standard ecological situations. Nonaka and Holme
[2007] considered “clumpiness” as a main characteristic of the food
distribution and developed an agent-based model for analysis of optimal
foraging. Grünbaum [1998] studied foraging in population of the ladybird
beetle (Coccinella septempunctata), while preying on the goldenrod aphid
(Uroleucon nigrotuberculatum). He used both experimental observation and PDE
models and analysed nonuniform “aggregated” distributions, in which foragers
accumulate at resource concentrations, and evaluated parameters of foragers’
strategy from experimental data.

From our point of view, the potential of PDE models is not exhausted despite
of growing popularity of the multiagent models in dynamics of space
distribution of populations [Rahmandad and Sterman, 2008]. Let us compare two
models:

  * 1.

A system of one PDE for population with kinesis and one ODE for substrate:

$\begin{split}&\partial_{t}u(t,x)=D\nabla\left(e^{-\alpha(as(t,x)-b)}\nabla
u\right)+(as(t,x)-b)u(t,x),\\\
&\partial_{t}s(t,x)=-gu(t,x)s(t,x)+d;\end{split}$ (4)

  * 2.

A system of one PDE for population with the constant diffusion coefficient
(i.e. without kinesis) and the same ODE for substrate:

$\begin{split}&\partial_{t}u(t,x)=D\nabla^{2}u+(as(t,x)-b)u(t,x),\\\
&\partial_{t}s(t,x)=-gu(t,x)s(t,x)+d.\end{split}$ (5)

These models are particular realisations of the system (1)

For the computations experiment, to solve partial differential equations,
first MATLAB pdpe [2017] function has been used for space dimension one. For
two-dimensional results below, the MATHEMATICA NDSolve [2014] solver with
Hermite method and Newton’s divided difference formula has been used.

We selected 1D benchmark (Fig. 1, compare to Fig. 1 in work of Grünbaum
[1998]) on the interval $[-50,50]$ with boundary conditions and with the
initial conditions:

$s(0,x)=Ae^{-\frac{x^{2}}{2}},u(0,x)=1,A=4.$

The values of the constants are: $D=10$, $\alpha=5$, $a=2$, $b=1$, $g=1$,
$d=1$.

It is the first expectation that the proper kinesis should improve the ability
of animals to survive in a clumpy landscape. We can see from Figs. 2, 3 that
the density burst for the system with kinesis is higher and the utilisation of
fluctuation of substrate goes faster than without kinesis.

Figure 2: Utilisation of a food patch. Population density burst and
relaxation: a) for animals with kinesis and b) for animals without kinesis.

a) b)

Figure 3: Utilisation of a food patch: a) dynamics of population density at
the centre of patch, b) dynamics of the food density at the centre of patch.

The fluctuation of food decreases faster for the system with kinesis. The
population density increases to a higher level for system with kinesis. This
is essentially non-linear effect because in the linear approximation near
uniform equilibrium models with kinesis (4) and without kinesis (5 coincide.

###  2.4 Utilisation of fluctuations in food density

For the second benchmark we consider fluctuations of substrate, which are
periodic in space and time. Our gedankenexperimet includes two populations of
animals. The only difference between them is that the first population
diffuses with kinesis (population density $v$), whereas the second (population
density $u$) just diffuses with the constant diffusion coefficient (no
kinesis). The equilibrium values of the diffusion coefficients coincide. These
populations interact by consuming the same resource as it is described by
equations LABEL:Eq:fluct below

$\begin{split}&\partial_{t}u(t,x)=D\nabla^{2}u+(as(t,x)-b)u(t,x);\\\
&\partial_{t}v(t,x)=D\nabla\left(e^{-\alpha(as(t,x)-b)}\nabla
v\right)+(as(t,x)-b)v(t,x);\\\
&\partial_{t}s(t,x)=-g(u+v)s+d[1+\delta\sin(w_{1}t)\sin(w_{2}x)],\end{split}$
(6)

with zero-flux boundary conditions and with the initial conditions:

$s(0,x)=0.5,u(0,x)=1,v(0,x)=1.$

The values of constants are: $D=10$, $\alpha=5$, $a=2$, $b=1$, $g=1$, $d=1$,
$w_{1}=w_{2}=1$.

Animals with kinesis have evolutionarily benefits in the explored non-
stationary condition. We observe extinction of the population without kinesis
(Figs. 4 and 5). This is the concurrent exclusion of the animals without
kinesis by the animals with kinesis. At the same time, the fluctuations of the
population with kinesis in space and time are lager then for the population
without it. This effect was expected: animals with kinesis rarely leave the
beneficial conditions an jump more often from the worse conditions. In the
conditions with the reproduction coefficient $r>0$ their density grows faster
and in the worse condition (with $r<0$) it decreases faster than for animals
with the constant diffusion coefficient.

Figure 4: Dynamics of population densities in fluctuating conditions: a)
growth of subpopulation with kinesis, b) extinction of subpopulation without
kinesis.  Figure 5: Dynamics of population densities in fluctuating conditions
at one point (the centre of the interval). Concurrent exclusion of the
population without kinesis by the population with kinesis.

###  2.5 Spreading of a population with the Allee effect

The reproduction coefficient of a population takes its maximal value at zero
density and monotonically decays with the density growth in the simplest
models of logistic growth and their closest generalisations. It is widely
recognised that such a monotonicity is an oversimplification: The reproduction
coefficient is not a monotonic function of the population density [Allee et
al., 1949, Odum and Barrett, 1971]. This is the so-called Allee effect. The
assumption of the negative growth rate for small values of the population
density is sometimes also included in the definition of the Allee effect. The
Allee effect is often linked to the low probability of finding a mate in a low
density population but non-monotonicity of dependence of the reproduction
coefficient on the population density and existence of the positive optimal
density can have many different reasons. For example, any form of cooperation
in combination with other density-dependent factors could also produce a non-
monotonic reproduction coefficient and existence of optimal population
density.

The simplest polynomial form of the reproduction coefficient with the Allee
effect is $r(u)=r_{0}(K-u)(u-\beta)$. A typical dependence $r(u)$ with the
Allee effect is presented in Fig. 6. The optimal density corresponds to the
maximal value of $r$ (by definition). The evolutionarily optimal strategy for
populations with the Allee effect is life in clumps with optimal density when
the average density is lower than the optimal density [Gorban and Sadovskiy,
1989]. This clumpiness appears even in homogeneous external conditions and is
the most clear manifestation of the Allee effect in ecology. There are
multiple dynamical consequences of the Allee effect [McCarthy, 1997, Bazykin,
1998]. In combination with diffusion it leads to a possibility of spread of
invasive species via formation, interaction and movement of separate patches
even in homogeneous external conditions [Petrovskii et al., 2002, Morozov et
al., 2006].

Figure 6: Reproduction coefficient with the Allee effect.

The reaction–diffusion equations for a single population with the Allee effect
in dimensionless variables are below for a system without kinesis (7) and for
a system with kinesis (2.5).

$\displaystyle\partial_{t}u(t,x)$ $\displaystyle=$ $\displaystyle
D\nabla(\nabla u)+(1-u)(u-\beta)u(t,x),$ (7) $\displaystyle\partial_{t}u(t,x)$
$\displaystyle=$ $\displaystyle D\nabla\left(e^{-\alpha(1-u)(u-\beta)}\nabla
u\right)$ $\displaystyle+(1-u)(u-\beta)u(t,x).$

The values of the constants are: $D=1$, $\alpha=10$, $\beta=0.2$.

We study invasion of a small, highly concentrated population into a
homogeneous environment. Equations (7) and (2.5) are solved for one space
variable $x\in[-50,50]$ with zero-flux boundary boundary conditions and with
the initial conditions:

$u(0,x)=Ae^{-\frac{x^{2}}{2}},\;A=1.$ (9)

b)

Figure 7: Evolution of a small, highly concentrated population with the Allee
effect from the Gaussian initial conditions (9) in a homogeneous environment:
(a) for animals with kinesis, (b) for animals without kinesis (7). The values
of constants are: $D=1$, $\alpha=5$, $\beta=0.2$.

a) b)

Figure 8: Evolution of a small, highly concentrated population with the Allee
effect from the Gaussian initial conditions (9) in a homogeneous environment:
(a) population density at the centre of the drop, (b) total population
dynamics (numerically integrated over space). The values of constants are:
$D=1$, $\alpha=5$, $\beta=0.2$.

The results of the numerical experiments (Figs. 7 and 8) demonstrate that
kinesis may delay invasion and spreading of species with the Allee effect. The
width of the cluster grows faster for the system without kinesis (Fig. 7), and
the total population dynamics numerically integrated over space (Fig. 8) also
demonstrates the faster growth of population without kinesis. The delay in
spreading appears because the animals with kinesis rarely leave dense
clusters, whereas animals without kinesis are spreading in areas with lower
values of the reproduction coefficient and populate them. this effect is also
reproduced in two-dimensional case presented in Figs. 10 and 11: an initial
Gaussian drop (Fig. 9) grows with kinesis (Fig. 10) slower than than without
kinesis (Fig. 11).

This effect of faster spreading could also lead to extinction for a population
with Allee effect. For small population density $u<\beta$ the reproduction
coefficient is negative. If diffusion is so fast that the local concentration
becomes lower than the threshold $\beta$ then the extinction of population
follows. In Fig. 13 we can see how the population without kinesis vanishes for
high diffusion, whereas the population with kinesis persists for the same
diffusion (Fig. 12) because it keeps low mobility at locations with high
reproduction coefficient.

Figure 9: Initial distribution ($t=0$): $u(0,x)=Ae^{-\frac{x^{2}+y^{2}}{2}}$,
$A=1$. Figure 10: 2D Allee effect with kinesis $D=0.2$, $\alpha=5$,
$\beta=0.1$, with Gaussian initial distribution (Fig. 9).  Figure 11: 2D Allee
effect without kinesis $D=0.2$, $\alpha=5$, $\beta=0.1$, with Gaussian initial
distribution (Fig. 9).  Figure 12: 2D Allee effect with kinesis, $D=0.5$,
$\alpha=5$, $\beta=0.1$, with Gaussian initial distribution (Fig. 9). Figure
13: 2D Allee effect without kinesis $D=0.5$, $\alpha=5$, $\beta=0.1$, with
Gaussian initial distribution (Fig. 9).

##  3 Discussion

We suggested a model of purposeful kinesis with the diffusion coefficient
directly dependent on the reproduction coefficient. This model is a
straightforward formalisation of the rule: “Let well enough alone”. The well-
being is measured by local and instant values of the reproduction coefficient.
Gorban et al. [2016] have discussed the problems of definition of instant
individual fitness in the context of physiological adaptation. Let us follow
here this analysis in brief. The proper Darwinian fitness is defined by the
long-time asymptotic of kinetics. It is non-local in time because it is the
average reproduction coefficient in a series of generations and does not
characterize an instant state of an individual organism [Haldane, 1932,
Maynard-Smith, 1982, Metz et al., 1992, Gorban, 2007]. The synthetic
evolutionary approach starts with the analysis of genetic variation and
studies the phenotypic effects of that variation on physiology. Then it goes
to the performance of organisms in the sequence of generations (with adequate
analysis of the environment) and, finally, it has to return to Darwinian
fitness. The ecologists and physiologists are focused, first of all, on the
observation of variation in individual performance [Pough, 1989]. In this
approach we have to measure the individual performance and then link it to the
Darwinian fitness. This link is not obvious. Moreover, the dependence between
the individual performance and the Darwinian fitness is not necessarily
monotone. (This observation was partially formalized in the theory of $r-$ and
$K-$ selection [MacArthur and Wilson, 1967, Pianka, 1970].) The notion
‘performance’ in ecology is ‘task–dependent’ [Wainwright, 1994] and refers to
an organism’s ability to carry out specific behaviours and tasks: to capture
prey, escape predation, obtain mates, etc. Direct instant measurement of
Darwinian fitness is impossible but it is possible to measure various instant
performances several times and treat them as the components of fitness in the
chain of generations. The relations between performance and lifetime fitness
are sketched on flow-chart (Fig. 14) following Wainwright [1994] with minor
changes. Darwinian fitness may be defined as the lifetime fitness averaged in
a sequence of generations.

Figure 14:  Flow diagram showing the paths through from genotype to Darwinian
fitness. Genotype in combination with environment determines the phenotype up
to some individual variations. Phenotype determines the limits of an
individual’s ability to perform day-to-day behavioural answer to main
ecological challenges (performances). Performance capacity interacts with the
given ecological environment and determines the resource use, which is the key
internal factor determining reproductive output and survival.

The instant individual fitness is the most local in time level in the
multiscale hierarchy of measures of fitness: instant individual fitness $\to$
individual life fitness $\to$ Darwinian fitness in the chain of generations.
The quantitative definition of the instant and local fitness is given by its
place in the equations. The change of the basic equation will cause the change
of the quantitative definition.

We have used the instant and local reproduction coefficient $r$ for defining
of purposeful kinesis. The analysis of several benchmark situations
demonstrates that, indeed, sometimes this formalisation works well. Assume
that this coefficient $r(u,s)$ is a monotonically decreasing function of $u$
for every given $s$ and monotonically increasing function of $s$ for any given
$u$. Then our benchmarks give us the following hints:

  * 1.

It the food exists in low-level uniform background concentration and in rare
(both in space and time) sporadic patches then purposeful kinesis defined by
the instant and local reproduction coefficient (2) is evolutionarily
beneficial and allows animals to utilise the food patches more intensively
(see Figs. 2 and 3);

  * 2.

If there are periodic (or almost periodic) fluctuations in space and time of
the food density $s$ then purposeful kinesis defined by the instant and local
reproduction coefficient (2) is evolutionarily beneficial and allows animals
to utilize these fluctuations more efficiently (see Figs. 4 and 5).

  * 3.

If the reproduction coefficient $r(u,s)$ is not a monotonically decreasing
function of $u$ for every given $s$ (the Allee effect) then the “Let well
enough alone” strategy may delay the spreading of population (see Figs. 7, 8,
10, and 11). This strategy can lead to the failure in the evolutionary game
when the colonization of new territories is an important part of evolutionary
success. This manifestation of the difference between the local optimisation
and the long-time evolutionary optimality is important for understanding of
the evolution of dispersal behaviour. At the same time, the “Let well enough
alone” strategy can prevent the effects of extinction caused by too fast
diffusion (see Figs. 12 and 13) and, thus, decrease the effect of harmful
diffusion described by Cosner [2014].

These results of exploratory numerical experiments should be reformulated and
transformed into rigorous theorems in the near future.

Purposeful kinesis is possible even for very simple organisms: it requires
only perception of local and instant information. For more complex organisms
perception of non-local information, memory and prediction ability are
possible and the kinesis should be combined with taxis. The idea of
evolutionary optimality can also be applied to taxis. This approach
immediately produces an advection flux, which is proportional to the gradient
of the reproduction coefficient. Cantrell et al. [2010] introduced and studied
such models. Moreover, the evolutionarily stable flux in these models should
be proportional to $u\nabla r$ [Averill et al., 2012, Gejji et al., 2012].
Here, $\nabla r$ could be considered as a ‘driving force’. It would be a very
interesting task to combine models of purposeful kinesis with these models of
taxis and analyse the evolutionarily stable dispersal strategies, which are
not necessarily unique even for one species [Buchi and Vuilleumier, 2012]. Of
course, the cost of mobility should be subtracted from the reproduction
coefficient for more detailed analysis.

If we go up the stair of organism complexity, more advanced effects should be
taken into account like collective behaviour and interaction of groups in
structured populations [Perc et al., 2013]. Moreover, the evolutionary
dynamics in more complex systems should not necessarily lead to an
evolutionarily stable strategy and cycles are possible [Gorban, 2007, Szolnoki
et al., 2014]. Even relatively simple examples demonstrate that evolutionary
dynamics can follow trajectories of an arbitrary dynamical system on the space
of strategies [Gorban, 1984]. Human behaviour can be modelled by differential
equations with use of statistical physics and evolutionary games [Perc et al.,
2017], but special care in model verifications and healthy scepticism in
interpretation of the results are needed to avoid oversimplification.

## Acknowledgement

We are very grateful to the anonymous referees. Their enthusiastic and
detailed comments helped us to improve the paper. AG and NÇ were supported by
the University of Leicester, UK. AG was supported by the Ministry of education
and science of Russia (Project No. 14.Y26.31.0022).

## References

  * Allee et al. [1949] Allee, W.C., Park, O., Emerson, A.E., Park, T., Schmidt, K.P., 1949. Principles of animal ecology. Saunders, Philadelphia.
  * Averill et al. [2012] Averill, I., Lou, Y., Munther, D., 2012. On several conjectures from evolution of dispersal. J. Biol. Dyn. 6(2), 117–130.
  * Gejji et al. [2012] Gejji, R., Lou, Y., Munther, D., Peyton, J., 2012. Evolutionary convergence to ideal free dispersal strategies and coexistence. Bull. Math. Biol. 74(2), 257–299.
  * Bazykin [1998] Bazykin, A.D., 1998. Nonlinear Dynamics of Interacting Populations. World Scientific, Singapore.
  * Buchi and Vuilleumier [2012] Büchi, L., Vuilleumier, S., 2012. Dispersal strategies, few dominating or many coexisting: the effect of environmental spatial structure and multiple sources of mortality. PloS one, 7(4), p.e34733.
  * Cantrell et al. [2010] Cantrell, R.S., Cosner, C. and Lou, Y., 2010. Evolution of dispersal and the ideal free distribution. Mathematical biosciences and engineering: MBE. 7(1), 17–36.
  * Chalub et al. [2004] Chalub, F.A., Markowich, P.A., Perthame, B., Schmeiser, C., 2004. Kinetic models for chemotaxis and their drift-diffusion limits. Monatshefte für Mathematik, 142(1–2), 123–141.
  * Chapman and Cowling [1970] Chapman, S., Cowling, T.G., 1970. The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases. Cambridge University Press.
  * Chen et al. [2010] Chen, W., Sun, H., Zhang, X. and Korošak, D., 2010. Anomalous diffusion modeling by fractal and fractional derivatives. Computers & Mathematics with Applications, 59(5), pp.1754-1758.
  * Cosner [2014] Cosner, C., 2014. Reaction-diffusion-advection models for the effects and evolution of dispersal. Discrete Contin. Dyn. Syst. 34, 1701–1745.
  * Dall et al. [2005] Dall, S.R., Giraldeau, L.A., Olsson, O., McNamara, J.M., Stephens, D.W., 2005. Information and its use by animals in evolutionary ecology. Trends Ecol. Evol. 20(4), 187–193.
  * Dobzhansky [1950] Dobzhansky, T., 1950. Evolution in the tropics, Am. Sci. 38 (2), 209–221.
  * Gorban [1984] Gorban A.N., 1984. Equilibrium Encircling. Nauka, Novosibirsk.
  * Gorban [2007] Gorban, A.N., 2007. Selection theorem for systems with inheritance. Math. Model. Nat. Phenom. 2(4), 1–45.
  * Gorban and Sadovskiy [1989] Gorban, A.N., Sadovskiy, M.G., 1989. Optimal strategies of spatial distribution: the Allee effect. Zh. Obshch. Biol. 50(1), 16–21.
  * Gorban et al. [2016] Gorban, A.N., Tyukina, T.A., Smirnova, E.V., Pokidysheva, L.I., 2016. Evolution of adaptation mechanisms: Adaptation energy, stress, and oscillating death. J. Theor. Biol. 405, 127–139.
  * Grünbaum [1998] Grünbaum, D., 1998. Using spatially explicit models to characterize foraging performance in heterogeneous landscapes. The American Naturalist, 151(2), 97–113.
  * Grünbaum [1999] Grünbaum, D., 1999. Advection–diffusion equations for generalized tactic searching behaviors. Journal of Mathematical Biology, 38(2), 169–194.
  * Hillen and Painter [2009] Hillen, T., Painter, K.J., 2009. A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58, 183–217.
  * Hofbauer and Sigmund [1998] Hofbauer, J., Sigmund, K., 1998. Evolutionary games and population dynamics. Cambridge University Press.
  * Haldane [1932] Haldane, J.B.S., 1932. The Causes of Evolution, Longmans Green, London.
  * Keller and Segel [1971] Keller, E.F., Segel, L.A., 1971. Model for chemotaxis. J. Theor. Biol. 30(2), 225–234.
  * MacArthur and Wilson [1967] MacArthur, R.H., Wilson, E.0., 1967. The Theory of Island Biogeography. Princeton Univ. Press, Princeton, N.J.
  * Maynard-Smith [1982] Maynard-Smith, J. 1982. Evolution and the Theory of Games, Cambridge University Press, Cambridge.
  * McCarthy [1997] McCarthy, M.A., 1997. The Allee effect, finding mates and theoretical models. Ecol. Model. 103(1), 99–102.
  * Méndez et al. [2012] Méndez, V., Campos, D., Pagonabarraga, I., Fedotov, S., 2012. Density-dependent dispersal and population aggregation patterns. J. Theor. Biol. 309, 113–120.
  * Méndez et al. [2010] Méndez, V., Fedotov, S. and Horsthemke, W., 2010. Reaction-transport systems: mesoscopic foundations, fronts, and spatial instabilities. Springer.
  * Metz et al. [1992] Metz, J.A., Nisbet, R.M., Geritz, S.A., 1992. How should we define ‘fitness’ for general ecological scenarios?. Trends. Ecol. Evol. 7(6), 198–202.
  * Morozov et al. [2006] Morozov, A., Petrovskii, S., Li, B.L., 2006. Spatiotemporal complexity of patchy invasion in a predator-prey system with the Allee effect. J. Theor. Biol. 238(1), 18–35.
  * NDSolve [2014] NDSolve. 2014. Wolfram Language & System Documentation Center. urlhttp://reference.wolfram.com/language/ref/NDSolve.html.
  * Nolting et al. [2015] Nolting, B.C., Hinkelman, T.M., Brassil, C.E., Tenhumberg, B., 2015. Composite random search strategies based on non-directional sensory cues. Ecol. Complex. 22, 126–138.
  * Nonaka and Holme [2007] Nonaka, E., Holme, P., 2007. Agent–based model approach to optimal foraging in heterogeneous landscapes: effects of patch clumpiness. Ecography 30(6), 777–788.
  * Odum and Barrett [1971] Odum, E.P., Barrett, G.W., 1971. Fundamentals of ecology. Saunders, Philadelphia.
  * Othmer et al. [1988] Othmer, H.G., Dunbar, S.R., Alt, W., 1988. Models of dispersal in biological systems. J. Math. Biol. 26(3), 263–298.
  * Othmer and Hillen [2000] Othmer, H.G., Hillen, T., 2000. The diffusion limit of transport equations derived from velocity-jump processes. SIAM J. Appl. Math. 61(3), 751–775.
  * Othmer and Hillen [2002] Othmer, H.G., Hillen, T., 2002. The diffusion limit of transport equations II: Chemotaxis equations. SIAM J. Appl. Math. 62(4), 1222–1250.
  * Parker and Smith [1990] Parker, G.A., Smith, J.M., 1990. Optimality theory in evolutionary biology. Nature 348(6296), 27–33.
  * Patlak [1953] Patlak, C. S. 1953. Random walk with persistence and external bias. B. Math. Biophys. 15, 311–338.
  * pdpe [2017] pdepe. 2017. Solve initial-boundary value problems for parabolic-elliptic PDEs in 1-D. MatWorks Documentation https://uk.mathworks.com/help/matlab/ref/pdepe.html.
  * Perc et al. [2013] Perc, M., Gómez-Gardeñes, J., Szolnoki, A., Floría, L.M. and Moreno, Y., 2013. Evolutionary dynamics of group interactions on structured populations: a review. Journal of the royal society interface, 10(80), p.20120997.
  * Perc et al. [2017] Perc, M., Jordan, J.J., Rand, D.G., Wang, Z., Boccaletti, S. and Szolnoki, A., 2017. Statistical physics of human cooperation. Physics Reports, 687, 1–51.
  * Petrovskii et al. [2002] Petrovskii, S.V., Morozov, A.Y., Venturino, E., 2002. Allee effect makes possible patchy invasion in a predator–prey system. Ecol. Lett. 5(3), 345–352.
  * Pianka [1970] Pianka, E.R., 1970. On $r$\- and $K$-selection, Am. Nat. 104 (940), 592–597.
  * Pough [1989] Pough, F.H., 1989. Performance and Darwinian Fitness: Approaches and Interpretations, Physiol. Zool. 62 (2), 199–236.
  * Rahmandad and Sterman [2008] Rahmandad, H., Sterman, J., 2008. Heterogeneity and network structure in the dynamics of diffusion: Comparing agent-based and differential equation models. Manage. Sci. 54(5), 998–1014.
  * Rosenblueth and Wiener [1950] Rosenblueth, A., Wiener, N., 1950. Purposeful and non-purposeful behavior. Philos. Sci. 17(4), 318–326.
  * Sadovskiy et al. [2009] Sadovskiy, M.G., Senashova, M.Y., Brychev, P.A., 2009. A model of optimal migration of locally informed beings. Dokl. Math. 80(1), 627–629.
  * Szolnoki et al. [2014] Szolnoki, A., Mobilia, M., Jiang, L.L., Szczesny, B., Rucklidge, A.M. and Perc, M., 2014. Cyclic dominance in evolutionary games: a review. Journal of the Royal Society Interface, 11(100), p.20140735.
  * Turchin [1989] Turchin, P., 1989. Population consequences of aggregative movement. J. Anim. Ecol. 58(1), 75–100.
  * Tyutyunov et al. [2017] Tyutyunov, Y.V., Titova, L.I., Senina, I.N., 2017. Prey-taxis destabilizes homogeneous stationary state in spatial Gause–Kolmogorov–type model for predator–prey system. Ecol. Complex. 31, 170–180.
  * Wainwright [1994] Wainwright, P.C., 1994. Functional morphology as a tool for ecological research. In P.C. Wainwright, S.M. Reilly (eds.), Ecological morphology: integrative organismal biology. University of Chicago Press, Chicago, 42–59.