A population changes model of spatial distributed two-species consocietum is considered. Species react
as a "predator-prey" system. Each species occupies two states. Migrations lead to the problem of a net
reproduction maximization. The basic model and its three reflexive modifications are considered

Kourshakova, Kristina A.

Senashova, Maria Yu.

Siberian Federal University Digital Repository

Journal of Siberian Federal University. Mathematics & Physics 2008, 1(3) 284-289УДК 517.55A model of Reflection Behaviour in Biological CommunitiesKristina A.Kourshakova∗Maria Yu.Senashova†Institute of computational modelling SB RAS,Akademgorodok, Krasnoyarsk, 660036,RussiaПолучена 05.02.2008, окончательный вариант 05.05.2008, принята 20.06.2008A population changes model of spatial distributed two-species consocietum is considered. Species reactas a "predator-prey" system. Each species occupies two states. Migrations lead to the problem of a netreproduction maximization. The basic model and its three reflexive modifications are considered.Keywords: population dynamics, purposeful migration, spatially distributed population.Models of spatially distributed populations and/or communities are still a matter of challengefor the students working in population biology, ecology, environmental sciences and mathematicalmodelling. The key problem in the present investigation is to construct an adequate model thatdescribes models and predicts the impact of the spatial structure on a community dynamics,as well as the migration processes. Adequate modelling of a spatial transfer of a being is thebasic difficulty here. Currently, the basic methodology addressing the problem is based on theimplementation of partial differential equation of “reaction÷ kinetics” type.Such an approach has a serious discrepancy. To be valid, the models strongly require that thebeings move over space randomly and spontaneously (aimlessly). This constraint is never met innature; even microorganisms control their spatial redistribution [1, 2, 3].Previously, there was proposed an approach to model the dynamics of a community withrespect to spatial effects based on the (micro)evolutionary principle [3, 4, 5]. In brief, the principleforces beings to migrate in the manner improving their existence. An improvement of existence ofbeings is the key question here; the answer is given by the net reproduction function [6, 7, 8, 5]k (ρ,−→r ). Here ρ = ρ(−→r ) is the (local) density of a population, and −→r is a point in space.Obviously, the population density ρ depends on the point at space.A net reproduction function results from two effects: the first is reproduction, and the secondis inheritance [6, 7, 8]. If these two effects take place, then an equation of the dynamics of abiological entity has the following form:ρ˙(−→r , t) = ρ(−→r , t) · k (ρ(−→r , t)) or ρt+1(−→r , t) = ρt(−→r , t) · k (ρt(−→r , t)) , (1)for discrete time. Here k(ρ) (k(N), respectively) is the net reproduction function. This functionmust be bounded from above. The equations (1) are the equations with inheritance. A compre-hensive theory of such equations, in the most general case, including the investigation of threetypes of evolutionary stability can be found in [5, 6, 7, 8]. The most general result is that thenet reproduction function k(ρ) must meet the extreme principle, for any free evolving biologicalcommunity. The function k(ρ) is maximal for those species (entities) that sustained during theevolution; k(ρ) = 0 in the case of continuous time, and k(ρ) = 1 for discrete time.∗e-mail: seriouskris@mail.ru†e-mail: msen@icm.krasn.ruc© Siberian Federal University. All rights reserved– 284 –Kristina A. Kourshakova, Maria Yu.Senashova Model of reflection behaviour in biological...Finally, the basic issue of our model of spatially dependent dynamics of a community is thatany migration must not be a random walk, but the result of the growth of a net reproduction.Further, we shall consider a model in discrete time and discrete space sites (called stations). Astudy of continuous model leads to severe technical problems, so we shall start with the discretecase.1. A Model of Two-Species CommunityWe shall study the dynamics of a community consisting of two species; they are supposed tobe “prey÷ predator” related. It means that one species exists due to external resources, but theother one lives due to the beings of the former species. It is also supposed that both speciesoccupy two stations∗ and migration means a transfer of beings (of any species) from station tostation. Any other movements (inevitable in a real situation) are neglected and supposed to haveno effect on the community dynamics.The dynamics of (isolated) subcommunity occupying a station is supposed to follow thediscrete analogue of classic Lotka-Volterra equation, if no migration takes place:Nt+1 = Nt · (a− bNt − fXt) , Mt+1 = Mt · (c− dMt − gYt) ,Xt+1 = Xt · (εfNt − hXt) , Yt+1 = Yt · (εgMt − kYt) . (2)Here Nt, Xt are the abundances of prey and predator, respectively, at the first station, andMt, Ytare the analogous variables at the second station. The parameters a and c determine the fertilityof the prey population, in the relevant stations; the parameters b and d describe the density-dependent self-regulation of this population, in the relevant station. The parameters h and kdescribe the similar density-dependent regulation at predator subpopulations. The parametersf and g describe, in the general form, an efficiency of the interaction of the beings of these twospecies, including the success of hunting, the success of escape, etc., in corresponding stations.Finally, ε represents the efficiency of conversion of the prey biomass into the predator biomass.1.1. A Basic Model of MigrationThe parameter p, 0 ≤ p ≤ 1 reflects the mobility of prey beings; similar, q, 0 ≤ q ≤ 1 is themobility of predator beings. These parameters are the transfer cost and might be interpretedas a probability of the successful migration from one station to the other; success here meansthat no damage for further reproduction has taken place. Migration from station A to station Bstarts, if living conditions “there” are better, than “here”, with respect to the transfer cost:(a− bNt − fXt) < p · (c− dMt − gYt) ,(εfNt − hXt) < q · (εgMt − kYt) , (3)for prey and predator beings, respectively. It should be stressed, that the migration act is executedindependently by each being, while the model considers it as a population event. The backwardmigration conditions are defined similarly:p · (a− bNt − fXt) > (c− dMt − gYt) ,q · (εfNt − hXt) > (εgMt − kYt) . (4)∗These are the sites considered together with the environmental conditions.– 285 –Kristina A. Kourshakova, Maria Yu.Senashova Model of reflection behaviour in biological...A migration act runs at each time moment t, independently for both species. If none of theinequalities (3, 4) is fulfilled, then no migration takes place, at the given time moment t. Preymigration flux ∆ (predator migration flux Θ, respectively) must equalize the inequalities (3, 4):(a− b(Nt −∆)− fXt) = p · (c− d(Mt + p∆)− gYt) ,(εfNt − h(Xt −Θ)) = q · (εgMt − k(Yt + qΘ)) (5a)in the case (3), orp · (a− b(Nt + p∆)− fXt) = (c− d(Mt −∆)− gYt) ,q · (εfNt − h(Xt + qΘ)) = (εgMt − k(Yt −Θ)) (5b)in the case (4). Then, ∆ (Θ, respectively) is equal to∆ =pc− a+ bN − pdM + fX − pgYb+ p2d, Θ =hX + εqgM − εfN − qkYh+ q2k(6a)for migration form station A to station B, and∆ =pa− c+ dM − pbN + gY − pfXd+ p2b, Θ =kY + εqfN − εgM − qhXk + q2h(6b)for the backward migration.Finally, let’s the outline how the basic model (2–6) works. For each time moment t, a di-rection and the migration fluxes (∆ and Θ, respectively) are determined. Then, the speciesredistribute themselves according to the Eqs. (6). Then, the abundances of the next generation{Nt+1, Xt+1; Mt+1, Yt+1} are determined, according to (2), with the relevant abundances of thecurrent generation {N˜t, X˜t; M˜t, Y˜t} defined by (5). If no migration takes place at the currenttime moment t, the stage with species redistribution is omitted.1.2. Reflexive BehaviourReflection in behaviour means the ability of a being to foresee and/or predict the behaviour of anopponent, in a competitive behavioural act. An implementation of reflexive behavioural strategyby animals is well known. Without going into further discussion of psychological or ethologicalaspects of the implementation of such strategies, let us concentrate on a simple model revealingtheir dynamic of effects.The basic model (2–6) does not exhibit any reflexive behaviour. An introduction of such abihaviour into the basic model may only be concerned with a spatial redistribution. In otherwords, a reflection of the optimal migration behaviour means that a being is able to “foresee” themigration behaviour of the competitive species. With respect to it, one may assume the followingpatterns of the reflection in the behaviour of the species: (i) preys reflect predators; (ii) predatorsreflect preys, and, finally, (iii) both species reflect each other.Thus, within the framework of our model, a reflection means that the species manifestinga reflection in the behaviour, detects the migration conditions and chooses the migration fluxaccording to the abundances of a competing species, that would be produced due to its migration,not the current ones. In the case (i) the formula for Θ will remain the same, but the formula for∆ must be replaced by∆ =pc− a+ bN − pdM + fX˜ − pgY˜b+ p2dorpa− c+ dM − pbN + gY˜ − pfX˜d+ p2b,(7)– 286 –Kristina A. Kourshakova, Maria Yu.Senashova Model of reflection behaviour in biological...in depending on the migration direction. Here X˜ and Y˜ are determined according to (5).Reciprocally, N˜ and M˜ are determined according to (5), in the case (ii), but the migrationflux Θ of the predator will be determined byΘ =hX + εqgM˜ − εfN˜ − qkYh+ q2korkY + εqfN˜ − εgM˜ − qhXk + q2h,(8)depending on the migration direction.Finally, if both species reciprocally reflect the behaviour of each other, then the basic modelshould be changed into the following one. On the first stage, both species determine the migrationfluxes according to basic model (5, 6). Then, they redefine the migration fluxes (and migrationdirection, as well as the fact of migration) so that each species changes the current abundancesof the competitive beings for those that could be produced due to a migration rule determinedby the basic model. So, they redefine the fluxes, redistribute themselves between the stations,and reproduce.2. Results and DiscussionThe purpose of this paper is to figure out the sets of the parameters providing an evolutionaryadvantage to a bearer of some (reflexive, or not) spatial distribution strategy. Evolutionaryadvantage here is understood as an excess of the total abundance of some species realizingreflexive strategy, in comparison to the same species in the case of realization of regular (non-reflexive) strategy.#1 a = 3.1, c = 1.49999, b = 0.00098, d = 0.00099, h = 0.00052, k = 0.0005, f = g = 0.00542,ε = 0.099, p = q = 0.99.#2 a = 2.5, c = 1.49999, b = 0.00098, d = 0.00099, h = k = 0.0005, f = g = 0.0054, ε = 0.099,p = q = 0.99.#3 a = c = 1.5, b = d = 0.0001, h = 0.0001, k = 0.0005, f = g = 0.0059, ε = 0.05, p = q =0.99.#4 a = c = 1.5, b = d = 0.0001, h = k = 0.00001, f = 0.0059, g = 0.0049, ε = 0.05,p = q = 0.99.#5 a = 1.7, c = 1.5, b = d = 0.0001, h = k = 0.00001, f = g = 0.005, ε = 0.05, p = q = 0.99.#6 a = 2.2, c = 2.1, b = d = 0.0001, h = k = 0.00001, f = g = 0.005, ε = 0.05, p = q = 0.99.First of all, it should be said, that the basic model (2 — 6) exhibits a great diversity ofthe limiting regimes. It may be a steady state (in both stations, for both species), limit cyclesof various length, and a complex irregular behaviour looking like a dynamical chaos. All thesepeculiar regimes may be met in combinations, with respect to a station and/of a species. Ingeneral, a decrease of transfer cost p and q yields a simplification of an observed regime. Both thebasic model (2–6) and its versions implementing various reflexive strategies of spatial distributionyield an expansion of the area of permissible parameter values, and the area of the phase space(i.e., abundance figures). All these issues are very interesting, from the point of view of the studyof the models of optimally migrating communities, but they fall beyond the scope of our research.– 287 –Kristina A. Kourshakova, Maria Yu.Senashova Model of reflection behaviour in biological...Table 1. Comparison of various strategies of space distribution. S — type of strategy: 1 —basic model, 2 — model i, 3 — model ii, 4 — model iii.S N X M Y S N X M YSet of parameter # 1 Set of parameter # 21 2089 5 4539 6 1 1513.5 1.4 1952.5 56.22 1930 20.5 2368 50 2 1438.8 4.1 2244.6 55.63 1692.9 43.7 1849.9 41.7 3 1524.5 0.47 1938.6 56.74 2142 0.1 5040 0.1 4 1530 0.01 5047 0.01Set of parameter # 3 Set of parameter # 41 3697 18 3698 18 1 3542 20.9 4073 14.32 4999 0.01 4999 0.01 2 4997 0.01 4998 0.013 3873 6.4 3869 6.5 3 3619 11.2 3739 84 4189 13.8 4190 13.8 4 4325 12.5 4384 11.5Set of parameter # 5 Set of parameter # 61 6994 0.05 4996 0.01 1 5.4 0.1 4.4 0.042 5688 32.1 4122 11.6 2 8406 80.5 7172 69.33 4664 19.5 2961 13.9 3 3.6 0.07 3.6 0.14 5412 30.5 4086 18.7 4 8404 80.5 7171 69.3What we did, was a comparative study of those four models (basic model and three versionswith reflexive behaviour) from the point of view of the evolutionary advantage. In fact, we triedvarious combinations of the parameters, similar for all four models, in order to identify the modelthat yields the highest total abundance of the species.To answer this question, we have carried out a series of computational experiments. Wehave calculated the abundance of each species, in each station, for four models with the sameparameter set. Then, the abundances of prey subpopulation (of predator subpopulation, in turn)have been added. Table 1 shows the results of this comparison. The parameters yielding theobserved regimes are shown below the Table.It is evident that there exist parameters sets yielding an evolutionary advantage for varioustypes of space distribution strategies. Yet, we did not study carefully the peculiarities of thelimit regimes relevant to each evolutionary advantageous situation. Meanwhile, one may expectthat the reflexive strategies provide an advantage for rather regular limit regimes, while thenon-reflexive strategy of space distribution is advantageous for chaotic-like, complicated limitregimes.The authors were supported by Grant NSh-3431.2008.9 from the President of Russian Feder-ation.References[1] Yu.L.Gurevich, N.S.Manukovsky, M.G.Sadovsky, Cell aggregation kinetics in continuouscultures, Dynamics of chemical & biological systems, Nauka, Novosibirsk, 1989, 159-168.[2] A.N.Gorban, M.G.Sadovsky, Population mechanisms of cell aggregation in continuous cul-tures, Biotechnology and Biotechnique, 2(1987), no. 5, 34.– 288 –Kristina A. Kourshakova, Maria Yu.Senashova Model of reflection behaviour in biological...[3] M.G.Sadovsky, Models of optimal migration of globally informed organisms, Mathematicalmodelling in biology & chemistry. Evolutionary approach, Novosibirsk, Nauka, 1991, 78-93.[4] S.A.Motolygin, D.A.Chukov, M.G.Sadovsky, Model of a population of optimally migratingorgamisms, Journal of Gen.Biology, 60(1999), 450.[5] M.G.Sadovsky, The simplest model of spatially distributed population with reasonablemigration of organisms, (arXiv q-bio.PE/0510004 2005)[6] A.N.Gorban, Systems with inheritance: dynamics of distributions with conservation ofsupport, natural selection and finite-dimensional asymptotics, (arXiv cond-mat/04054512005)[7] A.N.Gorban, R.G.Khlebopros, Darwin’s Demon: the Idea of Optimality and Natural Se-lection, Moscow, Nauka, 1988.[8] A.N.Gorban, Equilibrium encircling. Equations of chemical kinetics and their thermody-namic analysis, Nobosibirsk, Nauka, 1984.– 289 –

A model of Reflection Behaviour in Biological Communities

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