ADVANTAGE OR DISADVANTAGE OF MIGRATION IN A PREY-PREDATOR SYSTEM

Transcription

1 Far East Journal of Applied Mathematics 2015 ushpa ublishing House, Allahabad, India ublished Online: September Volume 93, Number 2, 2015, ages ISSN: ADVANTAGE OR DISADVANTAGE OF MIGRATION IN A RE-REDATOR SSTEM K. Sato 1, T. Hasegawa 2, S. Morita 1, J. oshimura 1,3,4 and K. Tainaka 1 1 Department of Systems Engineering Shiuoka University Hamamatsu , Japan sato@sys.eng.shiuoka.ac.jp morita@sys.eng.shiuoka.ac.jp jin@sys.eng.shiuoka.ac.jp tainaka@sys.eng.shiuoka.ac.jp 2 Graduate School of Information Sciences Tohoku University Sendai , Japan hasegawa@m.tohoku.ac.jp 3 Marine Biosystems Research Center Chiba University Uchiura, Amatsu-Kominato Chiba , Japan 4 Department of Environmental and Forest Biology State University of New ork College of Environmental Science and Forestry Syracuse, New ork 13210, U. S. A. Received: June 14, 2015; Revised: July 15, 2015; Accepted: July 27, Mathematics Subject Classification: 92D25, 92D40, 37N25. Keywords and phrases: pair approximation, prey-predator system, random walk, stochastic cellular automata. Communicated by K. K. Aad

2 110 K. Sato, T. Hasegawa, S. Morita, J. oshimura and K. Tainaka Abstract Migration or random walk in a prey-predator system is studied by pair approximation (A) theory and computer simulations. Both theory and simulations indicate that prey s migration is disadvantageous to prey. On the other hand, the effect of predator s migration is rather complicated. The migration of predators allows them to persist over a wide range of parameters in both the A theory and the simulation on a random network. However, it usually has negative effects on the persistence of predators in the simulation on a lattice. 1. Introduction opulation dynamics with interaction such as predator-prey or femalemale have now been studied from various points of view [1-4]. In real ecosystems with predator and prey, asymmetrical effect of migration is well known. Namely, predators more frequently move around compared to prey [5, 6]. So far, the migration process has been mainly modeled using random walk. Two main approaches are well known: diffusion-reaction equations [7, 8] and computer simulations on a lattice [9-12]. It is, however, difficult for these approaches to derive the asymmetrical effect of migration as discussed later. In the present paper, we apply the pair approximation (A) to a prey-predator system. For the sake of comparison, we carry out simulations both on a lattice and on a regular random network. It is known that the predictions of A theory almost agree with the simulation results on the random network [13, 14]. The A theory has been originally developed as a cluster variation approximation in physics [15]. Recently, it has been applied to stochastic spatial models for population or evolutionary dynamics [16, 17], such as the contact process (C) [18, 19] and the lattice Lotka-Volterra models [20-22]. The random walk on a lattice ecosystem has been first studied for C [11]. The A theory revealed that the steady-state density of the species increases with diffusion rate [11]. In ecology, C contains only a single species [23, 24]. We apply the A theory to a two-species (prey-predator) system to report the migration is advantageous (disadvantageous) for predator (prey).

3 Advantage or Disadvantage of Migration in a rey-predator System Model We consider a lattice or network, where each lattice site (network node) is labeled by (prey), (predator), or O (empty). Interactions are defined by + 2 (rate: 1), (1a) + O 2 (rate: r), (1b) O (rate: m), (1c) + O O + ( rate: D ), (1d) + O O + ( rate: D ). (1e) The processes (1a)-(1c) represent the standard prey-predator model [25, 26] first presented by Tainaka and Fukaawa (TF model) [27]: the reactions (1a), (1b) and (1c) denote the predation of, reproduction of and death of, respectively. Here r and m mean the reproduction rate of and the mortality rate of, respectively. The migration processes (1d) and (1e) are added to the TF model. The parameters D and D represent the migration (diffusion) rates of prey and predator, respectively. Note that we ignore the death process of prey, because we assume predator reproduces just after prey is eaten by predator. Computer simulation is performed on two networks: a square lattice and on regular random network. In the former, every site (node) interacts with four adjacent sites, while in the latter, each site randomly connects with just four sites [13, 14]. The initial condition is set to be random; three states O, and have the equal probability ( 1 3). Each network includes sites. The steady-state is obtained after Monte Carlo steps. The population dynamics of mean-field theory (MFT) is unchanged by migration. MFT for the system (1) agrees with that for TF model [28, 7]: = ro, & (2a) & (2b) = m,

4 112 K. Sato, T. Hasegawa, S. Morita, J. oshimura and K. Tainaka where O, and are the densities of empty sites O, preys and predators, respectively, ( O + + = 1). From equations (2a) and (2b), the densities of prey and predator in a stationary state (equilibrium) are given by ˆ ˆ 1 m = m, = r, (3) 1 + r where the hat indicates the equilibrium. The abundance of predator thus decreases with the increase of m. The predator survives for 0 m < mc ; the critical value is given by m C = 1 for MFT. When m 1, predator becomes extinct and all sites are occupied by prey. Here we consider the dynamics of pair densities σσ, which are the probabilities that a randomly chosen pair of neighboring sites are in states σ and σ for σ, σ, σ {,, 0}. We apply the pair approximation σσ σ σσ σ σ σ. We obtain five independent equations: & = ro, (4) & = m, (5) & = 2 r O + 2 r 2 O O D 2 O O O, (6) = + + O & O m D, (7) O 2 & = O r O O + m ( + ) OO O D O D D D, O (8)

5 Advantage or Disadvantage of Migration in a rey-predator System 113 where is the coordination number in the lattice and = 1. If we put D = D = 0 or if diffusion processes of both species are neglected, then the formulations (4)-(8) of pair approximation agree with the system obtained by Tainaka [29]. Hereafter we only consider an internal equilibrium for this closed system when it exists. The internal equilibrium solution follows the conditions: ˆ = m ˆ, (9) ˆ 1 ˆ m ˆ O = =, (10) r r ˆ ˆ ˆ ˆ r = O + ˆ ( D r), (11) + O ˆ 2 ˆ 2 ˆ 2 m O + m + D ˆ ˆ = O, (12) ˆ m + D O ˆ ˆ r + D 1 1 r m ˆ ˆ + r m + ˆ + = D r O O. (13) r + D + D rd ˆ ˆ Explicit values cannot be obtained, but instead we can numerically calculate O ˆ and ˆ corresponding to an internal equilibrium solution for arbitrary D and D using equations (9)-(13). If both diffusion rates are infinity ( D, D ), we have ˆ ˆ ˆ ˆ ˆ ˆ O O = =, ˆ ˆ ˆ ˆ ˆ ˆ O O = =. It follows that A theory corresponds to the MFT. Hence, the steady-state densities in the system with sufficiently large diffusion rates ( D 1, D 1) are predicted by MFT. In contrast, this does not hold, if a single species migrate ( D or D ).

6 114 K. Sato, T. Hasegawa, S. Morita, J. oshimura and K. Tainaka (a) (b) Figure 1. Results of Monte Carlo simulation on a square lattice ( r = 1). (a) and (b), respectively, represent the densities and in stationary state as a function of death rate (m) of. The predictions of mean-field theory (MFT) are also shown by straight lines.

7 Advantage or Disadvantage of Migration in a rey-predator System 115 (a) (b) Figure 2. Same as Figure 1, but the predictions of A theory (curves) and the results of simulation on a regular random network (plots) are shown.

8 116 K. Sato, T. Hasegawa, S. Morita, J. oshimura and K. Tainaka 3. Results The equilibrium densities are largely influenced by the values of diffusion constants D and D. Now, we report the results of the following three cases: (i) both species migrate equally ( D = D = 50), (ii) only preys migrate ( D 50 and D = 0) and (iii) only predators migrate = ( D = 0 and D = 50). In Figures 1 and 2, the results of simulations (plots) and prediction of A theory are illustrated. We confirm from Figure 2 that simulation results on the regular random network resemble the predictions of A theory. All results in Figures 1 and 2 indicate that the prey s migration is disadvantageous to prey: the values of decrease with the increase of D. On the other hand, the effects of predator s migration are largely different between Figures 1 (lattice) and 2 (A and random network) as follows: (1) Figure 1 (lattice): Increasing predator movement rate has complex effects. If prey is not mobile ( D = 0), then the critical mortality rate m C is slightly decreased by migration. Hence, the migration has a negative effect in the survival range of predator. However, if preys are highly mobile ( D = 50), then the migration allows predators to persist even with larger mortality rate. Similarly, the predator movement has a complicated effect on the steady-state (equilibrium) density of predators. When movement rate is increased, the density is higher or lower, depending on other parameters. (2) Figure 2 (A and random network): Increasing predator movement rate allows predators to persist over a wide variety of mortality rates. When D increases, m C is increased. Hence, the predator s migration helps the sustainability of predator. However, the effects on equilibrium predator density are still not always consistent: increasing movement rate drives equilibrium density up or down, depending on other parameters. 4. Conclusion and Discussions Above results may come from spatial structure of preys and predators. If

9 Advantage or Disadvantage of Migration in a rey-predator System 117 D = 0, then preys are locally clustered. It reduces the ability of predator to exploit preys, because preys in a cluster are safe. However, when preys are very mobile, the clusters of prey are dispersed. In this case, predators easily catch preys. Hence, when D increases, the density of prey (predator) decreases (increases). We discuss the reason why predator s migration is more advantageous on a random network, compared to a lattice. redators search preys by random walk. In the case of lattice, an individual frequently returns back to the same place; there are many loops in a lattice. In contrast, such loops are very rare in random network. The search is more effective on a random network compared to a lattice. Hence, the migration of predators allows them to persist over a wide range of parameters in both the A theory and the simulation on a random network. We discuss parameter sensitivity. We confirm the migration of prey is disadvantageous for prey: the steady-state densities always decrease with increasing D. On the other hand, the relation between and D is sensitively influenced by parameters. In Figure 3, the parameter dependence on the predator migration is summaried, where (a) and (b) are the results of A and lattice simulation, respectively. The red (blue) region denotes the case that is increased (decreased) by increasing of D. The yellow region means the extinction of predator. The A result (Figure 3(a)) reveals that increases in the vicinity of extinction boundary ( m = m C ). In other words, the survival area of predator is expanded by the predator s migration. On the contrary, the result of lattice simulation (Figure 3(b)) indicates that the migration makes the survival area narrow (decrease of m C ). It should be noted that even in the case of A, the density slightly decreases by predator migration, as shown in Figure 3(a). From the viewpoint of sustainability, whether the equilibrium density increases or decreases is not so important compared to whether the density is positive or ero. Next, we discuss the parameter sensitivity of D and D. For example, when D = 50 is replaced by D = 10, Figure 3(a) is almost unchanged, but the blue region in Figure 3(b) largely shrinks.

10 118 K. Sato, T. Hasegawa, S. Morita, J. oshimura and K. Tainaka (a) (b) Figure 3. The phase diagrams for the effect of predator migration. (a) and (b) are the results of A and lattice simulation, respectively. The vertical and horiontal axes mean the reproduction rate r of prey and death rate m of predator, respectively. The colors are determined by whether the predator density ( ) increases or not. We obtain in both cases D = 50 and D = 0, where D = 0 is always fixed. The red (blue) region denotes the case that the predator density ( ) is increased (decreased) by increasing of D The yellow region means the extinction of predator for both D = 50. and D = 0. It is not easy to derive the asymmetrical effect of migration by the approaches other than A theory. The reaction-diffusion equations have a serious problem. A typical reaction-diffusion system is obtained by modifying our mean-field theory: to equations (2a) and (2b), we, respectively, add Laplacian terms D Δ x and D Δ y. When we consider Neumann boundary condition or periodical boundary condition of a rectangular domain, then we can prove that there exists a spatially uniform steady-state for the above reaction-diffusion system which is independent of the diffusion constants D and D [30]. The theory of ESS (evolutionarily

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