Complex Population Dynamics in Heterogeneous Environments: Effects of Random and Directed Animal Movements

Int. J. Nonlinear Sci. Numer. Simul., Vol.13 (2012), pp. 299 309 Copyright 2012 De Gruyter. DOI 10.1515/ijnsns-2012-0115 Complex Population Dynamics in Heterogeneous Environments: Effects of Random and Directed Animal Movements Vikas Rai, 1 Ranjit Kumar Upadhyay 2; and Nilesh Kumar Thakur 2 1 Department of Mathematics, Faculty of Science, Jazan University, Jazan, Kingdom of Saudi Arabia 2 Department of Applied Mathematics, Indian School of Mines, Dhanbad, India Abstract. In this paper, we have investigated the complex dynamics of a one-dimensional spatial nonlinear coupled reaction-diffusion system with a Holling type IV functional response, akin to standard Michaelis-Menten inhibitory kinetics. Prey-taxis is included in a general reaction-diffusion equation to incorporate the active movement of predator species towards regions with high prey concentrations or if the predator is following some sort of cue (such as odor) to find the prey. We have carried out stability analysis of both the non-spatial model without diffusive spreading and of the spatial model. We performed extensive computer simulations to identify various parameter ranges for stable homogeneous solution. Our findings specifically elucidate the role of predator diffusion and prey-taxis in controlling emergent structures, and transitions towards spatiotemporal chaos. We observe that the increasing predator random movement and moderate value of prey-taxis stabilize the system. Keywords. Spatial plankton system, heterogeneous environment, prey-taxis, spatiotemporal pattern, directional movement. PACS (2010). 92B05, 92C16. 1 Introduction Reaction-diffusion equations have been the subject of intense research due to their rich variety of patterns. Conceptual predator- prey models have successfully been used to elucidate mechanisms of spatiotemporal pattern formation [1 3]. Wolpert [4] gave a clear and non-technical description of mechanisms of pattern formation in animals. * Corresponding author: Ranjit Kumar Upadhyay, Department of Applied Mathematics, Indian School of Mines, Dhanbad, India; E-mail: ranjit_ism@yahoo.com. Received: January 18, 2012. Accepted: March 19, 2012. Lee et al. [5, 6] investigate the necessary conditions for pattern formation in prey-taxis systems. They have also detected continuous travelling wave for prey-taxis in a model with Allee effect. One of the most efficient approaches for modeling the spatio-temporal dynamics of the interacting populations is based on the reaction-diffusion- advection equation [7]. The appearance of advection-driven heterogeneity in relation to multispecies interaction was studied by many authors [8 9]. Sapoukhina et al. [10] consider a reaction-diffusion-advection model for the dynamics of populations and investigated the role of prey-taxis in biological control. The advection term represents the movement of predator according to a basic prey-taxis assumption i.e., acceleration of predators is proportional to the prey density gradient. The predation process is divided into random movement described by diffusion and directed movement represented by prey-taxis. Random movement of plankton populations with different velocities can give rise to spatial patterns [11] and the directional movement of zooplankton plays a role in generating patterns in a plankton community model [12] due to the foraging behavior of zooplankton that move towards high phytoplankton density. Lewis [13] studied pattern formation in plant and herbivore dynamics and herbivore-taxis were seen to reduce the likelihood of pattern formation. The pattern formation in prey taxis models is still open to wide investigations. Recently, spatial heterogeneity of species has attracted much attention because it is closely related to the stability and coexistence of species in ecological systems. Two factors concern spatial heterogeneity as well as spatial pattern in which population distributed spatially and individuals interact locally. The first is internal noise which induce spatio-temporal pattern of species in concerning range. The second is predation intensity of species. Spatial heterogeneity and diffusion introduce qualitatively new types of behavior in predator-prey interaction [14]. Random diffusion alone does not usually explain well the movement of animals. Rapid dispersal of predators, modeled as random diffusion, has a stabilizing effect on community dynamics. Spatial heterogeneities [15 19] such as prey density gradients, give rise to prey-taxis. This phenomenon has been found to have a stabilizing effect on the dynamics [6] although these authors employ a different modeling approach. Many biological factors ought to alter the form of predator s functional response and thereby alter the dynamics of the predator and prey populations. The functional response

300 V. Rai, R. K. Upadhyay and N. K. Thakur encapsulates attributes of both the prey and predator biology. Hence, the handling time, search efficiency, encounter rate, prey escape ability, etc. should alter, in general, the functional responses [20]. Therefore, predator s functional response may be different when a particular predator preys different prey having different escape ability and in situations a prey is predated by different predators having different hunting ability. The structure of prey habitat is also responsible for changes in the functional responses. We have considered the Holling type IV functional response (also known as Monod Haldane function) which is similar to the Monod (i.e., the Michaelis-Menten) function for low concentrations but includes the inhibitory effect at high concentrations. For large value of predator s immunity from or tolerance of the prey the type IV functional response reduces to type II functional response. Therefore, a model could be more realistic from ecological point of view and interesting from mathematical point of view if one considers different predators functional responses and compares the dynamic effects of these functional responses on spatio temporal patterns. A prey-taxis model was derived by Kareiva and Odell [21], and they studied predator aggregation in high prey density areas. Later the model was applied to estimate the mean travel time of a predator to reach a prey resource [22]. The effect of prey-taxis on biological control and the formation of spatial patterns in two-spotted spider mites have been studied numerically by Chakraborty et al. [23 25]. It has been observed that for different values of prey-taxis the solutions are periodic, quasi-periodic and chaotic. In the present work, prey-taxis is included in a general reaction-diffusion equation to describe the active movement of predator species towards regions with high prey concentration or if the predator is following some sort of cue (such as odor) to find the prey [26]. Animals often use their sense of smell to locate food, identify mates and predators, and find suitable habitats. In both terrestrial and aquatic environments, the instantaneous temporal and spatial distribution of odors is complex and the predator is following some sort of odor to find the prey [27]. The recent work of Koehl [28] helps us in framing the model equations on the basis of empirical grounding. We extend the seminal work of Kareiva and Odell [21] to develop a theory of general movement mechanisms that can be used to address questions about if and how organism can achieve any other spatiotemporal distribution. We have investigated the contribution of prey and predator movements to spatial pattern formation. In particular, we consider foraging behavior of predators moving towards high prey density. We have assumed that the movement of predator is induced by the heterogeneity in the prey distribution and predator tends to aggregate in regions of high prey density. We have explored a new Lotka-Volterra type model system for spatio-temporal patterns and temporal dynamics in heterogeneous environment together with predator acceleration which is proportional to the prey density gradient. The predator s functional response is described by a standard Michaelis-Menten inhibitory kinetics i.e., Holling type IV functional response. Type IV functional responses have not been explored for their effect on pattern formation in reaction-diffusion -advection systems earlier. We specifically elucidate the role of predator diffusion and prey-taxis in controlling emergent structures, and transitions towards spatio-temporal chaos. 2 Model System We consider the dynamics of a general predator-prey model where at any location xand time t, the prey density is denoted by u.x; t/, that of predator by v.x; t/ and w.x; t/ denotes the velocity of predator. The directed component of the predator movement is described by prey-taxis according to the following assumption: the acceleration is proportional to the gradient of prey density. The random movement is represented by diffusion term. It is assumed that the variation of the predator velocity (i.e. acceleration) is determined by the prey density gradient, i.e., @ @t C w:r w D ru; where is the non-negative taxis coefficient. The interactions, e.g., intra-specific competition for space equalize the velocities of neighboring predators [29]. Thus, introducing diffusion in the predator velocity equation, above equation becomes @ @t C w:r w D ru C d 3 r 2 w; where d 3 is the non-negative diffusivity constant of the predator velocity. Since the velocity or its gradient, is sufficiently small, thus neglecting w:rw in the above equation, it becomes @w @t D @u @x C d @ 2 w 3 @x ; 2 where u.x; t/ can be thought of the prey density at the position x 2 Œ0; L at time t. w.x; t/ is the instantaneous velocity of the predator population movement defined at each spatial coordinate, is the taxis coefficient of the prey which represents the sensitivity of predators to heterogeneity of the prey density distribution. d 3 the diffusion coefficient for velocity of predator which is interpreted as an effect of social behavior: arrayal forces equalize speeds and directions of neighbors [30]. In the real world setting, above equations describe the gradual speeding up of the directed movement when a prey aggregate is being approached.

Complex Population Dynamics in Heterogeneous Environments 301 Our model that describes the dynamics of a predatorprey system with prey-taxis is the following reaction diffusion-advection system @u uv D u.1 u/ @t.u 2 = / C u C 1 C d @ 2 u 1 (1a) @x 2 @v @t D ˇ uv @.vw/ @ 2 v v C d.u 2 2 (1b) = / C u C 1 @x @x 2 @w D @u @t @x C d @ 2 w 3 (1c) @x 2 With zero flux boundary conditions wj xd0;l D @u ˇ D @v ˇ D 0: (2) @x ˇxD0;L @x ˇxD0;L The zero flux boundary condition (2) imply that no external input is imposed from outside. is the parameter measuring the ratio of the predator s immunity from or tolerance of the prey to the half-saturation constant in the absence of any inhibitory effect. At high prey densities, it becomes difficult for predators to identify and catch a prey because of excessive crowding. Half saturation constant is the prey density at which per capita predation rate becomes half its maximum. The predator s immunity results from group defenses or certain activities of the prey. ˇ is the parameter measuring the ratio of product of conversion coefficient with grazing rate to the product of intensity of competition among individuals of prey with carrying capacity, be the per capita predator death rate, d 1 and d 2 are diffusion coefficients of prey and predator. d 3 represents the diffusion coefficient for velocity of predator. 3 Non-spatial Model and its Stability Analysis In this section, we restrict ourselves to the stability analysis of the model system in the absence of diffusion in which only the interaction part of the model system is taken into account. We find the non-negative equilibrium states of the model system and discuss their stability properties with respect to variation of several parameters. We first analyze model System (1a) (1b) without diffusion. In this case, the model system reduces to the form du dt dv dt D ˇ D u.1 u/ uv.u 2 = / C u C 1 (3a) uv v (3b).u 2 = / C u C 1 The non-trivial points.u ;v / are given by u D S ps 2 4 =2; v D.1 u /.u 2 = / C u C 1 ; where S D Œ1.ˇ=/. The non-trivial points exist if S<0in the biologically meaningful domain u 0; v 0 under the constraints (i) ˇ>; (ii) Œ1.ˇ=/ 2 >.4= / and (iii) u <1: (4) The point E 0 is always unstable. E 1 is locally asymptotically stable in uv-plane provided the inequality ˇ=.1 C 2 / < is satisfied. If ˇ=.1 C 2 / >, then the equilibrium point E 1 is a saddle point with stable manifold locally in the u-direction and with unstable manifold locally in the v-direction or a saddle node if ˇ=.1 C 2 / D. In the following theorem, we present the necessary and sufficient conditions for the non-trivial positive equilibrium points E to be locally asymptotically stable. Denote, A D u 1 v. C 2u / ; (5)..u 2 = / C u C 1/ 2 B D ˇu v.1 u 2 = /..u 2 = / C u C 1/ : (6) 3 We present the results in the following Theorem. Theorem 1. The unique non-trivial positive equilibrium point E is locally asymptotically stable if and only if the following inequalities hold: (i) v.2u C / <.u 2 C u C / 2 ; (7) (ii) u 2 < : (8) The proof of this theorem follows from the Routh-Hurwitz criteria, and is omitted. The conditions in eq. (7) and (8) are obtained using the inequalities A>0and B>0. Applying the limit cycle theorem [31] in the model System (3a) (3b), yields the following conditions:..ˇ=/ 1/ 2 >4= and ˇ>: (9) Figure 1 shows the area in.1= /.ˇ=/ plane (as given by eq. (9)) which produces the stable limit cycle solutions. The parameter values for simulation experiments are derived from the shaded area. An oscillatory dynamics in model system for a typical set of parameter values D 0:3; ˇ D 2:33 and D 0:3 is presented in Figure 1. 4 Spatio-temporal Model and its Linear Stability Analysis In this section, we study the effect of diffusion on the model system about the interior equilibrium point. In order to derive the condition of stability for the point of

302 V. Rai, R. K. Upadhyay and N. K. Thakur Figure 1. Shaded region which produces stable limit cycle solutions (reproduced from Upadhyay et al. [32]), Time series displaying oscillatory dynamics in model System (3) for a typical set of parameter values drawn from the region marked by shaded area in Figure 1. The parameter values for which oscillatory dynamics was obtained are D 0:3, ˇ D 2:33 and D 0:3. equilibrium with prey-taxis and diffusion, we have considered the lineralized form of the model System (1) about E D.u ;v ;0/with small perturbations U.x;t/;V.x;t/ and W.x;t/ as follows: @U @ 2 U D a 11 U C a 12 V C d 1 @t @x 2 (10a) @V D a 21 U C a 22 V v @W @t @x C d @ 2 V 2 @x 2 (10b) @W D @U @t @x C d @ 2 W 3 @x : 2 (10c) where, a 11 D u 1 v 2 1C2u = = u 2 = Cu C1 ; a 12 D u = u 2 = Cu C1 ; 2 a 21 D ˇv 1 u 2 = = u 2 = Cu C1 ; a22 D 0: Let us assume the Fourier series solutions of system of eq. (10) of the form U.x;t/D X k U k exp.t/ cos kx; (11a) where p D k 2.d 1 C d 2 C d 3 / a 11 q D k 4.d 1 d 2 C d 2 d 3 C d 3 d 1 / k 2.d 2 C d 3 /a 11 a 12 a 21 r D k 2.d 1 d 2 d 3 k 4 a 11 d 2 d 3 k 2 a 12 a 21 d 3 a 12 v /: (13a) (13b) (13c) From Routh Hurwitz criteria, the stability of the equilibrium E in the presence of diffusion and taxis depend on the following conditions. p>0; r>0; pq r>0: (14a) (14b) (14c) From eqs. (12) (13) and using the Routh-Hurwitz criteria, the following theorem follows immediately. Theorem 2. If the conditions (7) and (8) are satisfied the positive equilibrium E is locally asymptotically stable in the presence of diffusion and taxis if and only if V.x;t/D X k W.x;t/D X k V k exp.t/ cos kx; W k exp.t/ sin kx: (11b) (11c) 0 <.u2 C u C / k 2 u v.s C pq/; (15) where s D k 2.a 12 a 21 d 3 C d 2 d 3 k 2.a 11 d 1 k 2 //. where k D n=l is the wave number for the mode n D 0; 1; 2;:::. The characteristic equation of the linearized system is given by 3 C p 2 C q C r D 0 (12) 5 Numerical Simulations The dynamics of the model System (1) is studied with the help of numerical simulation. The step lengths x

Complex Population Dynamics in Heterogeneous Environments 303 (c) (d) Figure 2. Space series generated for different values of prey-taxis D 0.05, 0.08, (c) 0.2, (d) 0.4 at fixed set of parameters values D 0:45, ˇ D 0:9, D 0:2, d 1 D 0:01, d 2 D 0:2, d 3 D 0:001. and t of the numerical grid are chosen sufficiently small so that the results are numerically stable. In this section we perform numerical simulations to illustrate the results obtained in previous sections. We choose the set of parameters D 0:45, ˇ D 0:9, D 0:2 for the model System (1). With the above set of parameters, we note that the positive equilibrium E exists, and it is given by.u ;v ;0/ D.0:375; 1:0546; 0:0/. The dynamics of the model System (1) is studied with the help of numerical simulation for one dimensional case. The plots (space vs. population densities) are obtained to study the spatial dynamics of the model systems. The temporal dynamics is studied by observing the effect of time on space vs. density plot of prey and predator populations. Spatiotemporal chaos is generated as a result of breaking the homogeneity and the formation of a non-stationary irregular spatial pattern when the local kinetics of the system is oscillatory for a wide class of initial conditions. In the absence of any spatial gradient, period-doubling bifurcations serve as the generating mechanism for chaos in these model systems. In order to observe the development of chaotic dynamics, we have chosen the value of prey-taxis coefficient greater than its critical value. The stability of the equilibrium point for the model System (1) under such condition is examined. We have studied the influence of prey-taxis parameter and diffusion coefficient d 2 for predator population. For the fixed values of parameters D 0:45, ˇ D 0:9, D 0:2, d 1 D 0:01, d 2 D 0:2, d 3 D 0:001, k D 3=5 the critical value of prey taxis parameter D 0:0741. In Figure 2, we have studied the spatial patterns generated by directed and random movement of predator population at fixed time t D 200. Figures 3 and 4 reflect the temporal dynamics of the model System (1). Finally Figures 5 and 6 show the spatiotemporal evolution of the model System (1). In Figure 2, we have studied the spatial pattern of model System (1) with increasing value of directed movement i.e.,

304 V. Rai, R. K. Upadhyay and N. K. Thakur (c) (d) Figure 3. Time series of model System (1) with D 0.05, 0.08, (c) 0.2, (d) 0.4 at fixed set of parameters values D 0:45, ˇ D 0:9, D 0:2, d 1 D 0:01, d 2 D 0:2, d 3 D 0:001. prey-taxis parameter, starting from D 0:05, belowthe critical value to the D 0:4, above the critical value of prey-taxis parameter. We observed that for < 0:0741 the system is stable in the presence of diffusion and prey-taxis and as we go above the critical value the system predicts instability towards a concentration density wave. In Figure 3, we have presented the time series of the model System (1) to observe the effect of prey-taxis starting from stable to unstable range of prey-taxis and for increasing value of predator random movement. The irregular spatial and temporal behaviors of the prey and predator densities of the model system are illustrated in first and second column. From Figure 3 we observed that for the increasing value of prey-taxis system appears to show the emergence of regular temporal oscillation. In Figure 4, we have studied the spatial activity for the model System (1a) (1c). As we increased the value of system size (i.e., x D 4 to 25) and plotted the corresponding time series and phase trajectories for the fixed set of parameters, we observed that the limit cycle changed into irregular temporal oscillations. In the homogeneous environment, population densities experience spatio-temporal chaos (Figures 5 and 6). Once again predator diffusion coefficient acts as a regularizing factor; but the predator species temporal evolution remains irregular even at higher values. The simulation study of the model system in the heterogeneous environment reveals that the prey-taxis and random movement of animal (i.e., predator population) affect the spatial distribution of species. Space-time plots of prey and predator densities display the irregular and complex nature of the dynamics of the model system. These spatiotemporal patterns reflect the effect of prey-taxis at both spatial and temporal scales, and at last we studied the spatial dynamics of the model systems for different value of diffusion coefficients d 2 D 0:1, 0.3, 0.5.

Complex Population Dynamics in Heterogeneous Environments 305 (c) Figure 4. Time series (first column) and phase trajectories (second column) of the model System (1) at the fixed set of parameters values D 0:45, ˇ D 0:9, D 0:2, d 1 D 0:01, d 2 D 0:1, d 3 D 0:001, D 0:2 for increasing value of system size x D 4, 10, (c) 25.

306 V. Rai, R. K. Upadhyay and N. K. Thakur (c) (d) Figure 5. Complex spatiotemporal patterns of predator density for model System (1) with D 0.05, 0.08, (c)0.2, (d) 0.4 at fixed set of parameters values D 0:45, ˇ D 0:9, D 0:2, d 1 D 0:01, d 2 D 0:2, d 3 D 0:001. In Figure 5, we have shown space-time plots of predator densities which display the irregular and complex nature of the dynamics of the model system with the increase of prey-taxis. These spatiotemporal patterns reflect the effect of environmental heterogeneity at both spatial and temporal scales. In Figure 6, we have studied the complex spatiotemporal patterns of the model systems for different values of diffusion coefficients of the predator population. The diffusion coefficient of predator has an influence over temporal evolution of species. Higher values of diffusion coefficient make the evolution of densities stationary. Lower values favour chaotic dynamics. Higher value of predator random movement is needed to transform irregular and chaotic dynamics into regular one. 6 Discussions and Conclusions Motivated by the competitive/cooperative dynamics of marine plankton, we have considered reaction-diffusionadvection equations for the general predator-prey interaction. We have studied the reaction-diffusion model with prey-taxis in one dimension and investigated its stability. The nontrivial equilibrium state E of prey- predator model is locally asymptotically stable in non-spatial as well as spatial case in the presence of prey-taxis under certain conditions. We observed that prey-taxis affect the predator s ability to maintain the prey density within some economic threshold. The expression in the right hand side of eq. (15) constitutes the critical prey taxis strength. We observed that

Complex Population Dynamics in Heterogeneous Environments 307 (c) Figure 6. Complex spatiotemporal patterns of prey density (first column) and predator density (second column) for model System (1) with d 2 D 0.1, 0.3, (c) 0.5 at fixed set of the parameters values D 0:45, ˇ D 0:9, D 0:2, D 0:2, d 1 D 0:01, d 3 D 0:001.

308 V. Rai, R. K. Upadhyay and N. K. Thakur for the fixed set of parameters the model System (1) is locally asymptotically stable for < 0:0741. Numericalsimulations suggest that for the moderate value of prey taxis (as D 0:08) the system is even stable. As we increase the value of prey-taxis ( D 0:2; 0:4) the stability is destroyed. Our results reflect that the prey taxis acts as a biological control. It has also been observed that the predator random movement acts as a regularizing factor. Very few reactiondiffusion models consider the effect of active animal movements i.e., movements which are executed by individuals of species in search of better food choices or to optimize the foraging gain (ratio of capture success to energy expenditure). Therefore, directed and random movement of animals acts as a controlling agent of the system. When the gradient of prey density increases, that of predator decreases in the opposite direction. Individuals of prey and predator move in groups. 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