ON PREDATOR-PREY POPULATION DYNAMICS UNDER STOCHASTIC SWITCHED LIVING CONDITIONS

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1 ON PREDATOR-PREY POPULATION DYNAMICS UNDER STOCHASTIC SWITCHED LIVING CONDITIONS Aleksandrs Gehsbargs, Vitalijs Krjacko Riga Technical University Abstract. The dynamical system theory has been extensively sed in the contemporary ecology. Sch theory is sed to describe biological systems and their main featres, to predict their behavior nder certain conditions, to find sitable explanations to biological phenomena, etc. [; ]. One of the advantages of the mathematical design is that models can depend only on a small nmber of parameters, still possessing capacity to describe biological systems adeqately [3]. There are several types of mathematical models that are sed. The most commonly applied type of models is differential and difference eqations that describe dynamics of poplations of given species [4]. Sch models can serve as powerfl tools to describe theoretical featres of dynamics of poplations. However, detailed data that allow estimation of parameters for sch models are not always available [3; 5]. Another type of models is developed via implementation of Markov chains that describe stochastic dynamics of poplations of given species [6; 7]. Parameters of sch models can be estimated based on censs data on the nmber of species that are sally more available [3]. Obviosly, sch models incorporate stochastic natre of life environments and stochastic natre of reglation processes [8]. Here we show how differential models can be extended to describe both nderlying deterministic poplation dynamics and stochastic transitions between the observed states. To perform or proposal approach we deal with the classic Lotka-Volterra eqation for the dynamics of predator-prey system analysis. Assming stochastic switching for some parameters we analyze this dynamical system as the ergodic Markov chain. Applying statistical approach jointly with MATHEMATICA, R, and MATLAB as the statistical software tools, we estimate the Markov transition probabilities and parameters of the steady-state stationary distribtion. Or mathematical design can be sed to explore both theoretical featres of mathematical models and their compliance with the real data. Keywords: dynamical systems, stochastic switching, Lotka-Volterra eqations. Introdction Dynamical system theory has been sccessflly applied to describe varios biological systems, to predict and explain their behavior [; 3]. Commonly sed mathematical models are deterministic difference and differential eqations. However, dring the past years both theoretical biologists and practitioners have claimed that deterministic models do not necessarily describe biological systems properly and frther investigation was needed [9-]. One of the obvios improvements is consideration of the stochastic natre of biological systems and their habitats [9; ]. Stochastic dynamics of biological systems has been observed in practice [8; 3] and varios approaches for modeling have been offered []. Usally sch models are assmed to be memoryless or markovian []. Markovian models are sed to describe stochastic transitions between the finite nmber of system s states. The advantage of sch models is that their parameters can be estimated based on censs data which are sally more available [3]. Moreover, stochastic switching between several states of biological system is observed in practice [3]. Here we describe how deterministic differential models can be extended to incorporate the stochastic natre of the biological systems and captre both nderlying deterministic dynamics and stochastic transitions between several observed states. We se classic predator-prey Lotka-Volterra eqations and incorporate stochastic switching of the model parameters. Lotka-Volterra eqations are a well-known model that was extensively stdied and improved [4], and sed to describe real-world data [5]. However, some criticism of deterministic Lotka-Voltera system was also proposed and frther improvements were offered [0; 6]. We analyze the system as the ergodic Markov chain and apply statistical modeling jointly in MATHEMATICA, R and MATLAB to estimate Markov transition probabilities and steady-state distribtions. Or mathematical design can be sed to explore both theoretical featres of mathematical models and their compliance with the real data. Model description Lotka-Volterra deterministic differential eqations are commonly sed to describe predator-prey, competition and other interactions between two poplations. Throghot the paper we se the following form of the eqations 4

2 dx dt dx dt = x ( a = x ( a b b x x b b x ), () x ) where x is the total nmber of predator species, x is the total nmber of prey species. Terms a and a describe propagation of species, terms b and b describe completion within same poplations, terms b and b describe interaction between two poplations. Conseqently, parameters a, a, b, b and b are chosen to be positive and parameter b is chosen to be negative. Deterministic Lotka-Volterra system has for eqilibrim states, of which we are interested in stable focs that corresponds to stable coexistence of two poplations: Fig.. Phase plot for deterministic Lotka-Volterra system with stable focs Location of the focs depends on the chosen parameters of the system and is given by the formla ab = b b ab = b b ab b b ab b b, () where we assme that b b b b 0. The parameters of the model that are sed in the formla have a definite meaning, i.e. a and a are the rates at which predator and prey species propagate, etc. These parameters are determined by internal and external factors for two poplations. Here we assme that external factors, sch as weather conditions or hman interference possess stochastic natre. For the sake of simplicity, we assme that external factors can be only in one of the two states, switching between these two states at random moments in time. That is, we assme that the parameters of the model switch between two following sets of vales: Table Two sets of possible vales of system parameters Set Set a = A, a = A, b = B, b = B, b = B, b = B a = C, a = C, b = D, b = D, b = D, b = D If at the first moment the system was described by Set, after some time it switches to Set, then again to Set and so on. Obviosly, between the switching moments the system is non-random, it is described by deterministic Lotke-Voltera model and has stable focs as one of the eqilibrim states. Ths, modeling sed is described by the following scheme: 43

3 Solve deterministic Lotka-Volterra eqation with parameters from Set Modeling scheme Step Step Step 3 Step 4 Change Change parameters from parameters from Set to Set Set to Set Solve deterministic Lotka-Volterra eqation with parameters from Set Table Fig.. Phase plot for different nmber of switching times (from left to right: 5 switchings, 6 switchings, 00 switchings) Fig. shows phase plots for increasing nmber of switching times (with same set vales). It can be clearly seen that eqilibrim state jmps between two points and system moves toward crrent eqilibrim state between the switching times. Time between switching is assmed to have exponential distribtion with parameter c: P c { t z} = e z >. (3) The average time between the moments of switching is eqal to c (inverse of the coefficient in the exponent). We assme that mean time c is large enogh so that the system has time to reach small epsilon region of the eqilibrim states. That is, c >> T where T is relaxation time of the system and the system spends most of the time near eqilibrim states. Definition of the Markov chain We define states of the Markov chain in the following manner. We choose two rectangle regions arond the eqilibrim states and define Markov chain states as the nmber of the region: Markov chain state=, if r Π Markov chain state=, if r Π Markov chain state= 3, else, (4) where r is a location of the system, Π is the first rectangle and Π is the second rectangle (Fig. 3). Fig. 3. Regions that define Markov chain states: region = area inside left rectangle; region = area inside right rectangle 44

4 Markov chain states are defined as the nmber of the region when the system is located at a stationary regime. That is, states of the Markov chain are defined only when the system reaches stationary mode. The idea behind sch definition is as follows. If the system with described switching is observed dring a long period of time, stationary regime shold be reached. We record locations of the system with time intervals eqal to c, assming that the system is observed in discrete moments in time. As a reslt of the modeling with switching moments we obtain the seqence of the Markov chain states, 3, 3,, 3,, 3,,,, 3,,, 3,... (5) The obtained seqence form the Markov chain properties of which can be stdied via modeling. Modeling procedre The system depends on the following. S) Parameters of the Lotka-Volterra system a, a, b, b, b and b. S) Rectangles Π, Π that define regions on phase plot and respective Markov chain states ( Fig. 3 ). S3) Average time c between switching (check formla 3) Modeling of the system depends on the following. S4) Vales chosen in S)-S3) S5) Time step vale dt sed for modeling (it shold be dt << c so that the system has time to reach epsilon-region of the eqilibrim state) S6) Nmber of switching moments N. As a reslt of modeling the seqence of the Markov chain states (formla 5) are obtained and can be sed to estimate transition probabilities: Probabilities are calclated as follows 3 P ˆ = ˆ p3. (6) ˆ ˆ 3 p3 p33 ˆ = nij p ; i, {,,3} n + n + n j ij, (7) i i where n ij is the total nmber of transactions from state i to state j in the seqence (5). In addition, empirical distribtions of x and x, and two-dimensional distribtion of (x,x ) at the switching moments can be obtained and investigated. Modeling reslts Here we present the modeling reslts. The modeling procedre is as follows. First we choose two sets of vales for system parameters, average time between the switching moments c, rectangles Π, Π, time step dt, nmber of the switching moments N. Next we generate N time intervals that have exponential distribtion and sm those p to get the total modeling time. Third, we simlate the system, switching parameters between sets as long as the time reaches the next switching interval. We assme that the system reaches stationary mode as long as half modeling time has passed. For the stationary regime we record Markov chain states and locations of the system with time intervals eqal to c, assming that the system is observed in discrete moments in time. We plot phase plot, histograms of x and x, Markov chain transition matrices and two-dimensional distribtion of (x, x ). Test. Parameters of the model are chosen to switch between two following sets of vales (all the parameters except a do not change) i3 45

5 Two sets of vales of parameters for Test Set Set a = 0., a = 0.4, b = 0., b = 0.4, b = 0.08, b = -0. a = 0., a = 0.8, b = 0., b = 0.4, b = 0.08, b = -0. Ths, location of the first eqilibrim focs (Formla, Set) and = 0.97 = 0.47 =.75 =.5 for the second eqilibrim focs (Set ). Other parameters where chosen as follows: c = 40; dt = 0.; N = 000; set.seed(0). Phase plot for Test is given in Fig. 4. Table 3 (8) (9) Fig. 4. Phase plot for Test (c=40, N=000) Empirical distribtions of the system locations are given by the following histograms are given in Fig. 5. Two-dimensional distribtion is given in Fig. 6. Fig. 5. Empirical distribtion of x (left) and x (right) Estimated matrix with transition probabilities is given by ˆP = (0)

6 Eigen vales are λ =.0, λ = 0.65 and λ 3 = As λ is the only vector with property λ = Markov chain is ergodic. Left eigenvector that corresponds to λ is and defines stationary distribtion. [ ] T xλ = 03 () Fig. 6. Two-dimensional distribtion for (x, x ) (c=40, N=000) Test. Parameters are chosen as for Test, except parameters c and N. Here we choose c = 80, N = 500. Phase plot for Test is given in Fig. 7. Fig. 7. Phase plot for Test (c=80, N=500) Empirical distribtions of the system locations are given by the following histograms (Fig. 8). Two-dimensional distribtion is given in Fig. 9. Fig. 8. Empirical distribtion of x (left) and x (right) at switching moments Estimated matrix with transition probabilities is given by ˆP = (0)

7 Eigen vales are λ =.0, λ = 0.54 and λ 3 = As λ is the only vector with property λ = Markov chain is ergodic. Left eigenvector that corresponds to λ is and defines stationary distribtion. [ ] T xλ = 0 () Fig. 9. Two-dimensional distribtion for (x, x ) (c=40, N=000) Reslts and discssion Here we describe and discss the reslts of modeling. We sed Lotke-Volterra predictor-prey eqations and modeled it with stochastic switching between the system parameters. Switching moments where assmed to have exponential distribtion. We selected two rectangle areas arond eqilibrim focses that defined Markov chain states. We stdied empirical distribtion of the system location and transition probabilities of the Markov chain. As we can see from Tests and, the obtained Markov chains were ergodic with stationary distribtions. Markov chain matrices with transitional probabilities can be sed as follows. There are two states of the dynamical system and it switches between these states at random. The states are defined as constraints on the nmber of species: the system is in the state / if the nmber of prey is from A to B and the nmber of predators is from C to D, where A, B, C and D are nmbers that define rectangles Π, Π. Analysis of the Markov chain can be sed to obtain transitional probabilities between given states and stationary distribtion. For both tests described here stationary distribtions show that the system spends approximately the same amont of time in each of the states. That is, approximately half of the time the system spends in state and half in state, with minor part of the time the system being observed in intermediate state 3. Empirical distribtions of x and x cannot be described with commonly sed theoretical distribtions bt possess some important graphical featres. Qalitatively there are hikes in empirical density arond eqilibrim states, assming that there are two small regions where the system stays most of the time. Assmption is confirmed by investigation of empirical two-dimensional distribtion of (x, x ). Conclsions. The modeling procedre shows that implementation of stochastic switching into the Lotke- Voltera predator-prey eqations leads to the ergodic Markov chain.. Empirical distribtions of the system location show that there are two regions arond eqilibrim states where the system is located most of the time. 3. Stationary distribtion of the obtained Markov chain shows that for the described tests approximately half of the time the system spends in one of the eqilibrim states. Ths, the system is likely to be observed in any of the eqilibrim states. References. Rama S.S., Uyenoyama M/K. The Evoltion of Poplation Biology; ISBN: , 009. Maynard Smith, J Models in ecology. Cambridge University Press, Cambridge. 48

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