Turing instabilities and spatio-temporal chaos in ratio-dependent Holling–Tanner model

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Abstract

In this paper we consider a modified spatiotemporal ecological system originating from the temporal Holling–Tanner model, by incorporating diffusion terms. The original ODE system is studied for the stability of coexisting homogeneous steady-states. The modified PDE system is investigated in detail with both numerical and analytical approaches. Both the Turing and non-Turing patterns are examined for some fixed parametric values and some interesting results have been obtained for the prey and predator populations. Numerical simulation shows that either prey or predator population do not converge to any stationary state at any future time when parameter values are taken in the Turing–Hopf domain. Prey and predator populations exhibit spatiotemporal chaos resulting from temporal oscillation of both the population and spatial instability. With help of numerical simulations we have shown that Turing–Hopf bifurcation leads to onset of spatio-temporal chaos when predator’s diffusivity is much higher compared to prey population. Our investigation reveals the fact that Hopf-bifurcation is essential for the onset of spatiotemporal chaos.

Highlights

► A spatio-temporal Holling–Tanner model with ratio-dependent functional response is considered. ► Turing and non-Turing patterns are obtained for specific choice of parameter values. ► Spatio-temporal chaos for parameter values within Turing–Hopf domain is reported. ► Various stationary and non-stationary patterns within and outside the Turing domain are reported.

Introduction

Predator–prey dynamics continues to draw interest from both applied mathematicians and ecologists as varieties of prey–predators type interactions are observed within terrestrial environment. Concerned prey–predator type models exhibit a wide range of interesting dynamic behaviors. While going for the mathematical modeling of predator–prey interaction, it seems to be simple initially but later it has been found that it possesses many challenging problems from the perspective of applied mathematics. The most challenging and crucial phase of modeling the population ecosystem is to examine and validate whether the concerned mathematical model can exhibit the proper behavior for the system under consideration.

From a theoretical aspect, the mechanisms that lead to these (often periodic) cycles were successfully explained by Lotka and Volterra in their now famous two-species predator–prey model which admit neutrally stable periodic solutions [36], [74]. In a more recent study, Hastings and Powell [23] examined the complex nonlinear behavior of three-species continuous-time ecological models and found them to be characterized by a far richer spectrum of dynamics than their well studied two species counterpart. In particular, they highlighted that three-species food web models are capable to exhibit chaotic oscillations and thus have relevance for the study of more complex population dynamics [68].

After the pioneering work of Alfred Lotka and Vito Volterra in the middle of 1920 for two species interactions, predator–prey models having prey-dependent functional response were studied extensively [16], [19], [40], [49]. Recently, there is a growing explicit search for biological and physiological evidences for the choice of appropriate functional response, especially when predators have to search for food (and therefore has to share or compete for food). Based upon experimental evidences and analysis of collected field data it was revealed that a suitable predator–prey interaction should be based on the so-called ratio-dependent theory. It states that the per capita predator growth rate should be a function of the ratio of prey to predator abundance, and hence is called ratio-dependent functional response. This is supported by numerous fields and laboratory experiments and observations [35] and the references there in. Arditi and Ginzburg [4] first proposed the ratio-dependent functional response for predator–prey system. Ratio-dependent functional responses are thought to be rare compared to prey-dependent functional response but ratio-dependent functional response is considered as a reasonable alternative to model the complicated interaction between prey and predators [1], [4]. Predator–prey model with ratio-dependent functional response have revealed a wide variety of rich dynamical behavior [12], [17], [26], [28], [32], [78]. Holling–Tanner prey–predator model has received significant attention from both theoretical and mathematical biologists [5], [25], [31], [57] although little work has been done with Holling–Tanner model with ratio-dependent functional response. In a recent paper Liang and Pan [35] have studied the local and global asymptotic stability of the coexisting equilibrium point, existence and uniqueness of Poincare–Andronov–Hopf-bifurcating periodic solution for the ratio-dependent Holling–Tanner model system. This work is extended by Saha and Chakrabarti [58] to a delay differential equation model. In this they studied the local and global stability of various equilibria along with the existence criteria for small amplitude periodic solution which bifurcates from coexisting equilibrium point. They have studied the behavior of solution trajectories near the degenerate equilibrium point (0, 0) and derived the analytical criteria for the shape trajectories approaching origin.

Segel and Jackson [60] first used reaction–diffusion system to explain pattern formation in ecological context based upon the seminal work by Turing [72]. Similar ideas were used to explain spatiotemporal pattern formation in case of plankton systems [34] and semiarid vegetation [30]. Numerous papers have been published in last three decades on spatiotemporal patterns produced by reaction–diffusion models of prey–predator interactions and other types of interacting ecological systems [3], [9], [14], [18], [27], [37], [38], [41], [42], [47], [48], [52], [53], [55], [56], [61], [64], [73]. Earlier works on pattern formation were focused to study the pattern formation arising from Turing instability. Some researchers have shown their keen interest in the study of the types of Turing-patterns, namely, cold spot pattern, hot spot pattern and labyrinthine pattern depending upon the choice of parameter values within Turing domain [10], [66], [67]. Recently the focus has shifted to study the non-Turing patterns those results in spatiotemporal chaos, wave of chaos and resulting pattern near codimension two Turing–Hopf bifurcation point [8], [10], [39]. It is an established fact that formation of spontaneous spatiotemporal pattern is an intrinsic characteristic of predator–prey interaction [51], [55], [62], [63].

Most of the literature on spatiotemporal pattern formation in prey–predator models dealt with prey-dependent functional response (see [10], [18], [38], [42], [55] and references cited therein), the study of models with ratio-dependent functional response are now getting attention from the researchers [3], [7], [8], [75]. Spatiotemporal pattern formation in case of Holling–Tanner type prey–predator models with ratio-dependent functional response still remains an interesting area of research.

In this present work, we have considered ratio-dependent Holling–Tanner model for predator–prey interaction where random movement of both species are taken into account. The basic model considered in this paper is a reaction–diffusion model where the reaction part follows ratio-dependent Holling–Tanner type interaction between prey and predator population and they are capable to diffuse over a two dimensional landscape. We have arranged our paper in the following manner: In Section 2, we have described Holling–Tanner predator–prey model with ratio-dependent functional response. In this section we have studied the conditions required for existence of interior equilibrium point, local asymptotic stability of interior equilibrium point, existence of small amplitude periodic solution arising from Poincare–Andronov–Hopf bifurcation and numerical investigation for existence of homoclinic orbit. In Section 3, we have obtained the conditions for the occurrence of Turing instability. Linear stability analysis for spatiotemporal model [50] around homogeneous steady-state is carried out and the conditions required for the onset of Turing-instability in terms of model parameters are derived. We have obtained the domain in two dimensional parametric space where spatially uniform steady-states for both prey and predator population become unstable under spatially inhomogeneous perturbations. We have performed extensive numerical simulations for parametric values in side and outside the Turing domain and obtained results are presented in Section 4. Choice of initial condition reflects small inhomogeneous perturbation from homogeneous coexisting steady state. Spatiotemporal chaotic pattern is reported for the parametric values lying outside the Turing domain. Finally we have summarized the main outcomes of present analysis in the concluding section.

Section snippets

Basic model: linear stability

Robert May developed a prey–predator model Holling type functional response [22], [24] to describe the predation rate and Leslie’s formulation [33], [40] to describe predator dynamics. This model is known as Holling–Tanner model for prey–predator interaction. Holling–Tanner model is capable to capture the essential features of real ecological systems namely mite/spider mite, lynx/hare, sparrow/sparrow hawk and some other species [13], [70], [77]. Holling–Tanner predator–prey model with

Spatiotemporal model: turing instability

In this section we consider the spatiotemporal Holling–Tanner model obtained form the temporal model (3a), (3b) by incorporating diffusion terms as followsnt=n(1-n)-npn+αp+2nx2+2ny2,pt=δpβ-pn+d2px2+2py2subjected to the no-flux boundary conditions and known positive initial distribution of populationsnν=pν=0,on(0,)×Ω,n(0,x,y)=n0(x,y)>0,p(0,x,y)=p0(x,y)>0for(x,y)Ω.Ωis the two-dimensional bounded connected square domain with boundary ∂Ω, /∂ν is the outward drawn normal

Numerical simulation: Turing and non-Turing pattern

We take fixed values α = .4 and β = 1.2 for numerical simulations and consider δ and d as controlling parameters. The spatiotemporal model (11a), (11b), (11c), (11d) becomesnt=n(1-n)-npn+.4p+2nx2+2ny2,pt=δp1.2-pn+d2px2+2py2.Numerical simulations of (15a), (15b) are carried out by transforming the infinite dimensional continuous model system to finite-dimensional form using discritization of time and space. Numerical simulations were carried out over a 200 × 200 lattice sites and spacing

Conclusion

In this paper we have studied a spatio-temporal prey–predator model where the interaction between prey and predator follows Holling–Tanner formulation with ratio-dependent functional response. The local asymptotic stability condition for coexisting equilibrium point and condition for Poincare–Andronov–Hopf-bifurcating periodic solution is described briefly. We have established the existence of homoclinic orbit through numerical investigation and identified the boundary for homoclinic

Acknowledgement

We highly appreciate Dr. Sergei Petrovskii, for providing valuable suggestions which helped us to improve the manuscript. We are also thankful to the anonymous referees for their careful reading and constructive comments which helped in better exposition of the paper.

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