Pattern Formation, Long-Term Transients, and the Turing Hopf Bifurcation in a Space- and Time-Discrete Predator Prey System

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1 Bull Math Biol DOI /s ORIGINAL ARTICLE Pattern Formation, Long-Term Transients, and the Turing Hopf Bifurcation in a Space- and Time-Discrete Predator Prey System Luiz Alberto Díaz Rodrigues Diomar Cristina Mistro Sergei Petrovskii Received: 27 October 2009 / Accepted: 4 October 2010 Society for Mathematical Biology 2010 Abstract Understanding of population dynamics in a fragmented habitat is an issue of considerable importance. A natural modelling framework for these systems is spatially discrete. In this paper, we consider a predator prey system that is discrete both in space and time, and is described by a Coupled Map Lattice (CML). The prey growth is assumed to be affected by a weak Allee effect and the predator dynamics includes intra-specific competition. We first reveal the bifurcation structure of the corresponding non-spatial system. We then obtain the conditions of diffusive instability on the lattice. In order to reveal the properties of the emerging patterns, we perform extensive numerical simulations. We pay a special attention to the system properties in a vicinity of the Turing Hopf bifurcation, which is widely regarded as a mechanism of pattern formation and spatiotemporal chaos in space-continuous systems. Counter-intuitively, we obtain that the spatial patterns arising in the CML are more typically stationary, even when the local dynamics is oscillatory. We also obtain that, for some parameter values, the system s dynamics is dominated by long-term transients, so that the asymptotical stationary pattern arises as a sudden transition between two different patterns. Finally, we argue that our findings may have important ecological implications. Keywords Predator prey system Allee effect Coupled map lattices Diffusive instability Pattern formation Turing Hopf bifurcation L.A.D. Rodrigues D.C. Mistro Departamento de Matemática, Universidade Federal de Santa Maria, Santa Maria, RS, Brasil S. Petrovskii ( ) Department of Mathematics, University of Leicester, Leicester, UK sp237@le.ac.uk

2 L.A.D. Rodrigues et al. 1 Introduction Pattern formation is one of the central issues in ecology (Levin 1992). Populations living in their natural environment are very rarely spatially homogeneous. Often this heterogeneity takes the form of a patchiness when areas with high population density alternate with areas where the given species is virtually absent. Sometimes, it can be linked to the properties of the environment. For instance, an insect population living in corn fields in considerable densities may be almost absent in the space between the fields simply because of unfavorable conditions there. There are, however, many cases when populations form large-amplitude spatial patterns without any apparent external forcing. A classical example of this behaviour is given by a plankton system where patches of high plankton density are commonly observed in a seemingly homogeneous environment (Fasham 1978; Martin 2003), but similar situations happen in terrestrial ecosystems as well (e.g. Greig-Smith 1979). Importantly, ecological patterning or patchiness has profound implications for population dynamics and ecosystem functioning (for instance, affecting the chances for species survival under adverse environment changes, e.g. see Allen et al. 1993; Petrovskii et al. 2004); thus understanding of this phenomenon is of crucial importance. In order to address the phenomenon of pattern formation mathematically, a number of approaches have been developed. The one that has probably been the most popular is given by diffusion-reaction equations. In particular, over the last three decades, there has been a significant progress in understanding pattern formation in predator prey and resource-consumer systems (Segel and Jackson 1972; Levin and Segel 1976, 1985; Pascual 1993; Sherratt et al. 1995, 1997; Lefever and Lejeune 1997; Petrovskii and Malchow 1999, 2001); for recent results and further reference, also see Malchow et al. (2008). An extension to diffusion-reaction equations is given by kernel-based equations (cf. Kot and Schaffer 1986; Neubert et al. 1995), which takes into account that the movement of individuals can follow scenarios more complicated than the standard diffusion. The above approaches are obviously based on the assumption that both the properties of the environment and the population densities can be considered as continuous functions of space. In reality, this does not always apply. For instance, consider a system of arable fields separated by non-agricultural areas with an insect species dwelling in the fields but not between them. Assume that each of the fields is not large so that each of the corresponding sub-population may be regarded as wellmixed, and hence described by a single number, i.e. its population size. A natural modelling framework for this system is spatially discrete. Insects can fly between the sites, and it means that individual fields are coupled by dispersal. Coupling can result in pattern formation when a few fields create a patch of either high or low population density, and the properties of the system as a whole the metapopulation such as its productivity, persistence etc., then depends on the details of the patchy structure. From the above example, the importance of good understanding of the dynamics of the spatially discrete systems is obvious. Meanwhile, while scenarios and mechanisms of pattern formation in spatially-continuous system are studied relatively well (e.g. see Okubo 1980; Meinhardt 1982; Grindrod 1996; Murray 1989; Okubo and Levin 2001; Malchow et al for results on diffusion-reaction systems and Kot

3 Pattern Formation, Long-Term Transients and Schaffer 1986; Kot et al. 1996; Andersen 1991; Neubert et al for the kernel-based systems), scenarios of pattern formation in a discrete system are not well understood yet. A type of model of particular interest is given by a Coupled Map Lattice (CML) where the dynamics is discrete both in space and time. The dynamics of CML has been a focus of research since mid-80s, first with regard to physical applications (Deissler 1984; Kaneko 1986, 1989; Brindley and Everson 1989) and later to ecology (e.g. Hassell et al. 1991; Cominsetal.1992; Allen et al. 1993; White and White 2005). There remain, however, a number of issues investigated rather poorly. In particular, a question about CMLs that has not been properly answered yet is under what conditions the emerging patterns are regular or irregular. In space-continuous systems, regular patterns are usually associated with the Turing instability (Turing 1952) when a positive steady state is stable locally (i.e. with respect to spatially homogeneous perturbations) but appears to be unstable with respect to inhomogeneous perturbation with a certain wave length. If the steady state becomes unstable due to the Hopf bifurcation, then the local kinetics becomes oscillatory. The point in the parameter space where the steady state loses its stability both to homogeneous and inhomogeneous perturbations is called the Turing Hopf bifurcation, and it is in vicinity of this point that the system is expected to exhibit irregular patterns and spatiotemporal chaos (Meixner et al. 1997; Baurmann et al. 2007). However, whether similar ideas can be applied to CMLs has remained unclear. In this paper, we consider a space- and time-discrete predator prey system described by a CML. We especially focus on the system s properties in vicinity of a discrete Turing Hopf bifurcation. We first provide a mathematically rigorous derivation of the conditions of the Turing instability. We then accomplish extensive numerical simulations and reveal a rich variety of spatiotemporal patterns. 2 Model The proposed model is a Coupled Map Lattice (CML) in which the state of the system, and consequently the dynamics, is described by a matrix, or equivalently, by attributing variable values to the nodes (which we call sites) of a two-dimensional integer lattice. We describe the state of the system considering only the population size in each lattice node and do not try to follow individuals in a micro-scale. The dynamics consists of two distinct stages: a dispersal stage and a reaction stage, the latter resulting from the inter- and intra-specific interactions (cf. Hassell et al. 1991). The dispersal stage is described by the equations N x,y,t = (1 μ N)N x,y,t + μ N 4 N z,w,t, (1) z,w V x,y P x,y,t = (1 μ P )P x,y,t + z,w V x,y μ P 4 P z,w,t, (2) where (N x,y,t,p x,y,t ) are the prey and predator densities at each node or site (x, y) = (i, j) at time t, (N x,y,t,p x,y,t ) are the densities after dispersal. The dispersal co-

4 L.A.D. Rodrigues et al. efficients μ N and μ P give the populations fractions of the prey and predator, respectively, that at each time step, leave their site to colonize the four nearest sites, V x,y ={(i 1, j), (i + 1, j), (i, j 1), (i, j + 1)} being the neighborhood of site (x, y). In the reaction stage, the predator prey interaction, which occurs locally in each node, is described by the equations N x,y,t+1 = f(n x,y,t,p x,y,t ), (3) P x,y,t+1 = g(n x,y,t,p x,y,t ), (4) where f and g represent the specific form of the interaction. 2.1 Stability of the Non-spatial System We assume that the local dynamics N t+1 = f(n t,p t ), P t+1 = g(n t,p t ), have a co-existence equilibrium point (N,P ). The criteria of local stability require that all eigenvalues of the Jacobian matrix A of the system have magnitude less than one. That takes place if and only if Tr A < 1+det A<2 (e.g. see Allen 2007). Correspondingly, the parameter range of the local stability is defined by the following three conditions: (i) Tr A<1 + det A, (ii) Tr A> 1 det A, and (iii) det A<1. If as a result of a change in parameter values, any of the conditions (i iii) is broken, the equilibrium looses its stability and a bifurcation occurs. In particular, if the condition (i) is broken, the eigenvalues are real and at least one of them is larger than 1. The corresponding bifurcation, which we will call the plus-one bifurcation, takes place for Tr A = 1 + det A. If the condition (ii) is broken, the eigenvalues are real and at least one of them is smaller than 1. This bifurcation, which we will call the minus-one bifurcation, takes place for Tr A = 1 det A. It is readily seen that it leads to formation of a two-point cycle (cf. Neubert et al. 1995). Finally, if the condition (iii) is broken, the eigenvalues are complex conjugate with the magnitude larger than 1. The corresponding Hopf bifurcation takes place for det A = 1.

5 Pattern Formation, Long-Term Transients 2.2 Local Population Dynamics Since a detailed investigation of the scenarios of pattern formation in the system (1 4) is hardly possible without extensive computer simulations, we now have to chose specific parametrization of the functions f and g describing the local population dynamics. In order to make our study as biologically relevant as possible, we chose f and g to ensure that the model would account for typical properties of the population dynamics observed in nature. Correspondingly, we consider the case that species N has overlapping generations so that, at any location in space, the population size at generation t + 1 consists of individuals that have just been born (with the linear growth rate r) and of the individuals that survived from the previous generation t: N t+1 = [ φ 1 (N t,r)+ φ 2 (N t ) ] N t φ 3 (P t ), where the number of offsprings per individual and the fraction of survived individuals are described by functions φ 1 and φ 2, respectively, and function φ 3 takes into account the impact of predation. We then assume that the maximum survival rate is reached for an intermediate value of the population size so that function φ 2 (N) has a unique maximum at a certain N>0. Mathematically, the simplest function that possesses this property is N φ 2 (N) = 1 + b 1 N 2, where b 1 > 0 is a parameter. Note that, since the maximum possible survival population fraction cannot exceed unity, b For the offsprings, we take into account intra-specific competition that results in the birth rate decreasing with an increase in the population size. Correspondingly, function φ 1 (N, r) can be written down as r φ 1 (N, r) = 1 + b 2 N 2, where b 2 > 0 is a parameter. Note that the factors affecting the survival rate do not necessarily coincide with the factors affecting the birth rate so that, generally speaking b 2 b 1. However, for the sake of simplicity, here we assume that b 2 = b 1 = b. As for the impact of predation, we consider a standard Nicholson Bailey-type parametrization using the exponential function, φ 3 (P ) = e P. Having taken all the above arguments together, we arrive at the following specific form for (3): N x,y,t+1 = (N x,y,t )2 + rn x,y,t 1 + b(n x,y,t) 2 exp( P x,y,t ). (5) The state variables N and P in (5) are scaled to dimensionless values (using a standard procedure which we do not show here for the sake of brevity), and that accounts for the absence of parameters in the exponent and in front of N 2 in the numerator.

6 L.A.D. Rodrigues et al. It is readily seen that, in the absence of the predator, the prey population described by (5) exhibits the Allee effect when, at low population densities, the per capita growth rate increases along with an increase in the population density. That agrees very well with our intention to keep the model biologically realistic as the Allee effect was proved to be very common in population dynamics (Stephens and Sutherland 1999; Courchamp et al. 1999, 2008). The Allee effect can be weak or strong. The strong Allee effect introduces a population threshold; the population must surpass this threshold to grow, below it the species goes to extinction. In the weak Allee effect, there is no extinction threshold (the extinction state is unstable) but the population shows a small growth factor for low population densities. In (5), strong and weak Allee effects for the prey population can be taken into account by considering r<1orr>1, respectively. In this paper, we will concentrate our attention on the weak Allee effect which has been less studied, at least in discrete models. For the predator dynamics, we consider the case when the predator is a generalist so that it can survive in the absence of prey N (which means that there is an alternative source of food, not included into the model) but the species N is the predator s preferable prey, so that the predator growth rate may become (significantly) larger in the presence of species N: P t+1 = ψ 1 (P t ) ψ 2 (N t ) P t, where ψ 2 is an increasing function. With regard to the predator s inherent dynamics described by function ψ 1, we assume that the growth rate s density dependence exhibits overcompensation which is caused by a direct interference between the individual predators. The rate of decay in ψ 1 (P ) at large predator density should then depend on P 2 rather than on P. Correspondingly, we describe the predator local dynamics at position (x, y) as P x,y,t+1 = P x,t exp[ m(p x,y,t )2] exp(cn x,y,t ), (6) where c and m are positive parameters; m quantifies the intensity of intra-specific competition. It is readily seen that a positive equilibrium (N,P ) of the system (5 6)isgiven by the following equations: N = m c (P ) 2, P = ln N + r 1 + b(n ) 2. (7) A closer look at (7) shows that, depending on the parameters, the system can have between one and three equilibria. It is not possible to obtain an analytical condition to distinguish between these cases. However, (7) can be solved numerically for different parameter values. The result are shown in Fig. 1. Therefore, apart from a relatively small domain, the system (5 6) has only one steady state. For that reason and also in order to avoid unnecessary complexity, in the below we restrict out study to the case of a unique equilibrium.

7 Pattern Formation, Long-Term Transients Fig. 1 Regions in the parameter space where the system (7)has either one or three positive equilibria. When crossing a border of the three-equilibria domain, two of the steady states merge and disappear; the thick line therefore corresponds to a non-robust case when the system has two steady states Fig. 2 A bifurcation diagram with b as the controlling parameter; other parameters m = 2, c = 2andr = 3 At the steady state (N,P ), the Jacobian reduces to ( ) rb(n ) 2 +2N +r N A = (1+b(N ) 2 )(N +r) cp 1 2m(P ) 2. (8) The form of the community matrix (8) (in particular, the bulky expression for the element a 11 ) makes it hardly possible to obtain the conditions of the equilibrium stability explicitly. Correspondingly, it does not seem possible to follow the bifurcation structure of the system analytically. In order to make an insight into these issues, we performed numerical simulations. A typical bifurcation diagram is shown in Fig. 2 and the structure of the system s parameter space is shown in Fig. 3. We observed that the co-existence equilibrium is stable in a wide range of parameter values. When it loses stability, the dynamics becomes oscillatory; in particular, we have found simple two-point cycles and multi-periodic n-point cycles. For some other parameter values, we have also observed quasi-periodicity and chaos. 2.3 Diffusive Instability Conditions The results of the previous sections, if interpreted in terms of the CML, address the system stability with respect to a spatially homogeneous perturbation. Now we are

8 L.A.D. Rodrigues et al. Fig. 3 Regions in the parameter space where different types of dynamics are obtained for model (5 6): (a) Local (non-spatial) stability of the coexistence state is found in region B. In the strips A, we have found n-point cycles, and in region C extinction of predators is observed. (b) RegionA corresponds to quasi-periodic dynamics, region B corresponds to local stability, region C corresponds to predator extinction. Points marked as I, II,...,VII correspond to parameter Sets I, II,...,VII, respectively; see details in the text. (c) Region A corresponds to 2-point cycles, B corresponds to the stability of the coexistence equilibrium, C corresponds to extinction of predators, D corresponds to n-point cycles and chaos. The boundaries of all the domains are obtained numerically going to consider what can be the conditions of the system stability with respect to a small spatially heterogeneous perturbation: N x,y,t = N + ε x,y,t, P x,y,t = P + δ x,y,t, (9)

9 Pattern Formation, Long-Term Transients where (N,P ) is the steady state of the non-spatial system and ε x,y,t and δ x,y,t are small numbers, generally speaking, different at different lattice nodes. Our aim is to investigate whether these perturbations grow or decay in the course of time when the dispersal is taken into account. Plugging (9) into(1 4), we obtain N x,y,t = N + ε x,y,t = N + (1 μ N )ε x,y,t P x,y,t = P + δ x,y,t = P + (1 μ P )δ x,y,t + μ N 4 (ε x 1,y,t + ε x+1,y,t + ε x,y 1,t + ε x,y+1,t ), (10) for the dispersal stage, and + μ P 4 (δ x 1,y,t + δ x+1,y,t + δ x,y 1,t + δ x,y+1,t ), (11) N x,y,t+1 = N + ε x,y,t+1 = f(n + ε x,y,t,p + δ x,y,t ), (12) P x,y,t+1 = P + δ x,y,t+1 = g(n + ε x,y,t,p + δ x,y,t ), (13) for the reaction stage. Using the Taylor series expansion for f and g about (N,P ), and retaining only the linear terms with respect to δ and ɛ, from (10 13) we obtain ( ε x,y,t+1 = a 11 (1 μ N )ε x,y,t + μ ) N 4 (ε x 1,y,t + ε x+1,y,t + ε x,y 1,t + ε x,y+1,t ) + a 12 ( (1 μ P )δ x,y,t + μ P 4 (δ x 1,y,t + δ x+1,y,t + δ x,y 1,t + δ x,y+1,t ) (14) ( δ x,y,t+1 = a 21 (1 μ N )ε x,y,t + μ ) N 4 (ε x 1,y,t + ε x+1,y,t + ε x,y 1,t + ε x,y+1,t ) + a 22 ( (1 μ P )δ x,y,t + μ P 4 (δ x 1,y,t + δ x+1,y,t + δ x,y 1,t + δ x,y+1,t ) where a ij are the elements of the Jacobian; see (8). The Fourier-type generic solution of the system of difference equations is [ ] [ ] εx,y,t α1 = λ t cos q δ x,y,t α 1 x cos q 2 y, (16) 2 where λ, α 1, α 2, q 1, and q 2 are unknown. By substituting (16) into(14 15) and imposing the constraint that the eigenvalues of the resulting matrix have a magnitude less than one, we find the following conditions for the stability of the homogeneous steady state: k N a 11 + k P a 22 < 1 + k N k P det A, ), ), (15)

10 L.A.D. Rodrigues et al. where k N k N (z 1,z 2 ) = 1 (1 cos z 1 cos z 2 )μ N and k P k P (z 1,z 2 ) = 1 (1 cos z 1 cos z 2 )μ P are called characteristic functions, z 1 = q 1+q 2 2 and z 2 = q 1 q 2 2. The desired result is then achieved by adding the conditions that the local equilibrium must be stable. The necessary and sufficient conditions for the diffusive instability on a CML then become as follows: Tr A < 1 + det A, (17) det A<1, (18) k N a 11 + k P a 22 > 1 + k N k P det A. (19) A dispersal-driven plus-one bifurcation occurs when inequalities (17) and (18) hold, and (19) turns into 1 (k N a 11 + k P a 22 ) + k N k P det A<0. (20) Correspondingly, a diffusion-driven minus-one bifurcation occurs when inequalities (17) and (18) hold, but (19) turns into 1 + (k N a 11 + k P a 22 ) + k N k P det A<0. (21) Interestingly, the above conditions for the Turing instability through either plusone or minus-one bifurcation coincide with those obtained by Neubert et al. (1995) for a kernel-based integrodifference equation. It indicates that there is an inherent relation between CMLs and the kernel-based models. Indeed, White and White (2005) attempted to establish a direct link between the two models by considering the dispersal kernel as a sum of the Dirac delta-functions with singularities placed at the lattice nodes. It is readily seen that the integro-difference equation then turns into a CML straightforwardly. However, here we argue that this approach is not mathematically rigorous. Linking a continuous space system to a discrete lattice implies that both the dispersal kernel and the population densities are discretized, the latter then becoming a composition of delta-functions as well. The integrand then turns into a sum of squared delta-functions, i.e. an object that is mathematically ill-defined. We want to emphasize that, in this paper, the above conditions of the Turing instability are obtained straightforwardly by applying relevant linear stability analysis to a CML, without assuming any relation between the space-discrete system and the space-continuous one. Therefore, here these conditions are obtained rigorously for the first time, even that the relation between the two types of models may have seemed to be intuitively clear. Note that the instability conditions determine restrictions on the elements of the community matrix as well as on the movement parameters. For the sake of brevity, we include this results here without calculations; all details can be found in Neubert et al. (1995). It appears that for diffusive instability the inequality a 12 a 21 < 0 is necessary, which means that the interspecific interaction must be of prey-predator or activatorinhibitor type. It is also required that (a 11 1)(a 22 1)<0, (22)

11 Pattern Formation, Long-Term Transients provided both characteristic functions are positive at the wave number where a plusone bifurcation is to occur. Similarly, in the case of the minus-one bifurcation, it is necessary that (a )(a )<0, (23) provided both characteristic functions are positive at the wave number where the bifurcation is to occur. On the other hand, when k N > 0 and k P < 0, conditions (22) and (23) are replaced by a 11 > 1 or a 22 < 1, and vice versa. If the dynamical parameters are in the domain of diffusive instability then the movement parameters must be chosen properly. The plus-one bifurcation correspondingto(20) can be written as a quadratic inequality with respect to cos z 1 cos z 2 : where Q(z 1,z 2 ) = α(cos z 1 cos z 2 ) 2 + β cos z 1 cos z 2 + γ < 0, (24) α = μ N μ P det A, β = (1 μ N )μ P det A + (1 μ P )μ N det A μ N a 11 μ P a 22, γ = 1 (1 μ N )a 11 (1 μ P )a 22 + (1 μ N )(1 μ P ) det A. Obviously, the point (z 1,z 2 ) where Q(z 1,z 2 ) touches the (z 1,z 2 ) plane must satisfy the following conditions: Q(z1,z 2 ) = 0, (25) Q (z1 z,z 2 ) = 0, 1 (26) Q (z1 z,z 2 ) = 0. 2 (27) It is readily seen that (25) results in cos z 1 cos z 2 = β ± β 2 4αγ 2α Equation (26) holds if and only if. (28) sin z 1 = 0 or cos z 2 = 0 or cos z 1 cos z 2 = β 2α. (29) In its turn, (27) holds if and only if sin z 2 = 0 or cos z 1 = 0 or cos z 1 cos z 2 = β 2α. (30)

12 L.A.D. Rodrigues et al. In order to obtain the relations describing the boundary of the Turing domain in the parameter space, we now need to consider different cases described by (28 30). 1. Consider β 2α < 1, then conditions (29 30) reduce to cos z 1 cos z 2 = β 2α. Having compared it to (28), we arrive at β 2 4αγ = 0. (31) 2. Consider then cos z 1 cos z 2 = β 2α given by β 2α > 1, is not possible and the boundary of the Turing domain is sin z 1 = 0 or cos z 2 = 0 and sin z 2 = 0 or cos z 1 = 0, which appears (omitting standard calculations for the sake of brevity) to be equivalent to (a) α β + γ = 0 or (b) γ = 0. (32) Therefore, the boundary of the Turing domain corresponding to plus-one bifurcation is described by relations (31 32). For a minus-one bifurcation, the analysis is similar. The conditions of minus-one bifurcation corresponding to (21) can also be written as a quadratic inequality with respect to cos z 1 cos z 2 : where Q(z 1,z 2 ) = α(cos z 1 cos z 2 ) 2 + β cos z 1 cos z 2 + γ < 0, (33) α = μ N μ P det A, β = (1 μ N )μ P det A + (1 μ P )μ N det A + μ N a 11 + μ P a 22, γ = 1 + (1 μ N )a 11 + (1 μ P )a 22 + (1 μ N )(1 μ P ) det A. Hence, the same criteria (31) and (32) hold, up to the obvious change β to β and γ to γ. Interestingly, however, for all the parameters that we have checked, it appears that β/(2α) > 1, therefore, condition (31) has no effect on the minusone parameter domain.

13 Pattern Formation, Long-Term Transients 3 Simulations Our goal in this section is to examine the existence of complex spatial patterns in the system (1 6) in a vicinity of the Turing Hopf bifurcation. Different types of stability/instability is expected to result in different dynamical regimes. In order to check that we performed extensive numerical simulations. Simulations were run in a two-dimensional habitat with reflective zero-flux boundary conditions. Extensions to other boundary conditions are straightforward and appear to produce little effect (cf. Hassell et al. 1991). For the initial conditions, we considered a small random perturbation of the uniform steady state of the system. In order to visualize simulation results, we use two kinds of graphics: the density plots, where the dark (light) colors correspond to high (low) concentrations, and the time series of the population size on the global scale (i.e. its total over the whole domain) and on a local scale (in a single site). We applied conditions (31 32) in order to obtain the value of the dispersal coefficients where diffusive instability takes place; see Fig. 4. The region marked as +1 gives μ N and μ P corresponding to a plus-one bifurcation values, for which the solution is expected to describe a spatial pattern that becomes stationary in the large-time limit. The region marked as 1 gives μ N and μ P corresponding to minus-one bifurcation, for which the solution is expected to describe a spatiotemporal pattern that is not only heterogeneous in space but also oscillating in time. These theoretical predictions are in a good agreement with simulations. Here, we want to mention that the system (5 6) depends on four parameters and, therefore, its full numerical investigation in the whole parameter space (i.e. varying all the parameters in a sufficiently wide range) is hardly possible. Instead, we have applied a different approach. We fixed two of the parameters at some hypothetical values, r = 1.2 and b = 0.5. We then considered a few combinations of the other two re- Fig. 4 The structure of (μ N,μ P ) parameter plane as given by the Turing instability conditions for (a)the plus-one bifurcation (dashed and solid curves correspond to conditions (31) and(32), respectively) and (b) the minus-one bifurcation

14 L.A.D. Rodrigues et al. action parameters c and m. Since in this paper we are especially interested in the scenarios of pattern formation in vicinity of the Turing Hopf bifurcation, all these combinations were chosen to be either close to or inside the parameter domain where the system s local dynamics is oscillatory, cf. domain A in Fig. 3b. Specifically, we considered the following three parameter sets: Set I: r = 1.2, b = 0.5, c = 0.5 and m = 0.5. Set II: r = 1.2, b = 0.5, c = 0.2 and m = 0.5. Set III: r = 1.2, b = 0.5, c = 0.2 and m = 0.2. For each of these sets, in order to reveal the impact of different types of the diffusion instability, we consider different values of the dispersal coefficients μ N and μ P. Figures 5 to 9 show the results obtained for the parameter Set I (which correspond to a point in region B in Fig. 3b). We first make simulations for μ N = and μ P = Figures 5 and 6 show, respectively, the prey and predator spatial distribution obtained at different time. In the large-time limit (see the bottom row of the figures) the population distribution converges to a stationary irregular pattern that is made of small-size patches or spikes. Note that while the prey is only present inside the spikes, the predators are distributed more or less over the whole habitat, although in the regions with high prey densities predators are more abundant. This quantitative difference between the distributions of prey and predator must obviously be attributed to the large value of the dispersal coefficients ratio, μ P /μ N A closer look at the patterns in Figs. 5 and 6 reveals a curious feature of the system dynamics. The first stage of the system dynamics leads to formation of a smooth pattern; see Figs. 5c and 6c obtained for t = Then from this smooth pattern, the spike pattern appears almost suddenly, over a (relatively) short time interval. It indicates that the system possesses an intrinsic time scale (being on the order of 10 3 for these parameter values) that can be associated with long-living transients. The existence of this long memory of the system is seen even better from the graphs of the total population size against time as shown in Fig. 7. The population sizes remain approximately constant for t up to about After that, the predator total size converge relatively fast to a new value. Interestingly, although during this transition time the prey spatial distribution changes considerably (cf. Figs. 5c and 5d), its total size remains almost the same. A qualitatively different dynamical behaviour is observed for the dispersal coefficients corresponding to the minus-one bifurcation. Figures 8 and 9 show the results obtained for μ N = and μ P = 0.5. In this case, the initial populations distribution promptly converges to a regular checkboard-like spatial pattern. Once the pattern is formed, the system dynamics does not undergo any qualitative changes. The total population size remains almost constant (see Fig. 9a), exhibiting only a very slight increase at about t = 2000; therefore, the effects of long-term transients are not prominent in this case. The pattern is periodical both in space and time: the local population size oscillates according to a simple two-point cycle (Fig. 9b). The system exhibits different dynamics when parameter values are chosen closer to the Hopf bifurcation domain. Figures 10 to 13 are obtained for parameter Set II which is still inside region B but lies very close to the border of region A; see point

15 Pattern Formation, Long-Term Transients Fig. 5 Prey spatial distribution shown at time t = 1, t = 50, t = 1000, t = 2000, t = 3000, and t = 5000, (a) to(f), respectively, for reaction parameters r = 1.2, b = 0.5, c = 0.5, and m = 0.5 (Set I) and dispersal parameters μ N = and μ P = 0.99

16 L.A.D. Rodrigues et al. Fig. 6 Predator spatial distribution shown at time t = 1, t = 50, t = 1000, t = 2000, t = 3000, and t = 5000, (a) to(f), respectively; parameters are the same as in Fig. 5

17 Pattern Formation, Long-Term Transients Fig. 7 Total prey (continuous curve) and predator (dashed curve) population vs. time for the same parameter values as in Figs. 5 and 6 Fig. 8 (a) Prey and(b) predator density in the habitat after four thousand time-steps for the reaction parameters as in Set I and the dispersal rates μ N = and μ P = 0.5 II in Fig. 3b. In particular, Fig. 10 shows the parameter domains in the (μ N,μ P ) parameter plane where the system can exhibit dispersal-driven instability for this value of reaction parameters. Figure 11 shows the prey spatial distributions at different times obtained for μ N = 0.1 and μ P = 0.95, which corresponds to a plus-one bifurcation. The spatial distribution of predator (not shown here for the sake of brevity) exhibits similar features. Now, instead of a spike-like patchy spatial pattern obtained above (see Figs. 5 and 6), the initial conditions evolve, in a large-time limit, to a stationary labyrinthinelike pattern. No long-living transients are observed in this case and, once the effect of the specific form of the initial conditions is forgotten (which occurs on the time scale of t 100), the system s dynamics does not undergo any qualitative change. Figure 12 shows the prey and predator spatial distribution obtained after five thousand time-steps for μ N = 0.98 and μ P = 0.05, that is, in the region of the dispersal parameters corresponding to a minus-one bifurcation. As expected for the minus-one bifurcation, diffusion-driven instability does not lead to a stationary pattern. In each

18 L.A.D. Rodrigues et al. Fig. 9 (a) Total prey (continuous curve) and predator (dashed curve) population size vs. time. (b) Local (one node) prey (solid curve) and predator (dashed curve) population size vs. time. The parameters are the same as in Fig. 8 Fig. 10 Regions in the dispersal parameter plane (μ N,μ P ) where diffusion-driven instability can occur for the reaction parameters Set II through (a) the plus-one bifurcation (dashed and solid curves correspond to conditions (31)and(32), respectively) and (b) the minus-one bifurcation site, the populations oscillate according to a simple two point cycle (Fig. 13a). The total populations remain constant, though (Fig. 13b). Interestingly, for this parameter values the spatial patterns exhibit two-periodicity in space. While the short-range spatial oscillations take place between any two neighboring sites (which results in a checkboard-like structure), the pattern appears to be periodical on the global scale with the period L/2 where L is the total length of the system; see the light-grey stripe in the middle of domain in Fig. 12. The above sets of reaction parameters correspond to the case when the steady state is locally (non-spatially) stable. Now we are going to consider a change in the system dynamics when the steady state becomes locally unstable, which results in oscillatory dynamics. Especially when local oscillations are combined with the conditions for the Turing instability that can be regarded as a discrete analogue of the Turing Hopf bifurcation one can expect formation of complex non-stationary spa-

19 Pattern Formation, Long-Term Transients Fig. 11 Prey spatial distribution shown at time t = 1, t = 50, t = 100, t = 200, t = 9000, and t = 10000, (a)to(f), respectively, for the reaction parameters as given by Set II and the dispersal parameters μ N = 0.1 and μ P = 0.95

20 L.A.D. Rodrigues et al. Fig. 12 Spatial distribution of prey (a) and predator (b) att = 5000 for the reaction parameters give by Set II and dispersal coefficients μ N = 0.98 and μ P = 0.05 Fig. 13 (a) Local (one node) population of prey (solid line) and predator (dashed line) and(b) the total population of prey (solid line) and predator (dashed line) for the parameters the same as in Fig. 12 tiotemporal patterns. However, rather counter-intuitively, simulations show that in the spatially extended system this is not necessarily the case; instead, the system can converge to a stationary pattern. Figures 14 to 19 shows the results obtained for parameter Set III, which corresponds to a point in the parameter region A in Fig. 3b. The corresponding dynamics of the non-spatial system appears to be quasi-periodic; see Fig. 14. A choice of dispersal parameter values corresponding to a plus-one bifurcation (cf. Fig. 15) appears to stabilize the system dynamics. Figure 16 (obtained for μ N = 0.01 and μ P = 0.95) shows how the oscillations in the total population size, being of very large amplitude at the first stage of the system dynamics (up to about t 500), then gradually decay, so that the population sizes are constant for large t. The corresponding spatial population distribution (see Fig. 17) is distinctly irregular in space and stationary in the large-time limit.

21 Pattern Formation, Long-Term Transients Fig. 14 III (a) Phase plane and (b) local predator population size in the non-spatial system for parameter Set Fig. 15 Regions in the dispersal parameter plane (μ N,μ P ) where diffusion-driven instability can occur through (a) the plus-one bifurcation (dashed and solid curves correspond to conditions (31) and(32), respectively) and (b) the minus-one bifurcation. The dynamical parameters are r = 1.2, b = 0.5, c = 0.2, and m = 0.2 (Set III) However, for a minus-one bifurcation, the same Set III of parameter values results in a different spatiotemporal dynamics. In this case, the total population size exhibits persistent oscillations (Fig. 18) with a large period, while the spatial distribution forms a two-scale pattern where the species are present (forming a checkboard structure) only inside a patch of a certain size but appears be absent outside of this patch; see Fig. 19. Although our main goal was to reveal the patterns in the vicinity of the Turing Hopf bifurcation, in order to have a broader understanding of the system s properties we have also performed simulations for other parameter values. In particular, we have considered the following parameter sets: Set IV: r = 1.2, b = 0.5, c = 0.17, and m = Set V: r = 1.2, b = 0.5, c = 0.35, and m = 1.5.

22 L.A.D. Rodrigues et al. Fig. 16 Total population of prey (solid curve) and predator (dashed curve) versus time for parameter Set III in the spatially extended system (CML) in the case of the plus-one bifurcation, μ N = 0.01 and μ P = 0.95 Fig. 17 (a) Preyand(b) predator spatial distribution after a heterogeneous spatial equilibrium is reached in the large-time limit (t = 2000). Parameters are the same as in Fig. 16 Set VI: r = 1.2, b = 0.5, c = 1.5, and m = 1.5. Set VII: r = 1.2, b = 0.5, c = 1.7, and m = 0.5. The dispersal parameters were chosen corresponding to the Turing instability through the plus-one bifurcation. The stationary spatial distribution of prey obtained in the large time limit is shown in Fig. 20 (the distribution of predator exhibits similar features and hence is not shown here for the sake of brevity). The system dynamics considered in a wider parameter range therefore exhibits a rich variety of patterns. For a small value of c (set IV), the system forms a cold spots pattern; see Fig. 20a. On the contrary, large values of c (sets VI and VII) results in hot spots patterns where the density of the hot spots (i.e. the domains where the prey density is high) is rather low for a small value of m but becomes much higher for a larger m; see Figs. 20c and 20d, respectively. For an intermediate value of c (set V), we observe a dislocation pattern when a checkboard-type structure (which is stationary in this case) is distorted along certain broken lines.

23 Pattern Formation, Long-Term Transients Fig. 18 Total population of prey (solid curve) and predator (dashed curve) versus time for parameter Set III in the spatially extended system in the case of the minus-one bifurcation, μ N = and μ P = 0.01 Fig. 19 (a) Preyand(b) predator spatial distribution at t = Parameters are the same as in Fig Local and Global Stability of the Spatial Equilibrium Conditions (17 19) and (20) of the Turing instability define the region(s) in the dispersal parameters plane where the steady spatially homogeneous population distribution becomes unstable with respect to small heterogeneous perturbations. As a result of this instability, the system is driven away from the equilibrium and forms patterns, which in the case of the plus-one bifurcation, appear to be stationary in the largetime limit. Thinking about this situation in term of dynamical systems, it means that, under the conditions of Turing instability there is a spatially heterogeneous attractor which is a stationary solution of (1 4). Now, an interesting question is how this situation may change if the system parameters are chosen outside of the Turing domain. The spatially homogeneous steady state solution then becomes stable with respect to small perturbations of any wave length, but does it mean that the heterogeneous attractor disappear? We hypothesize that, at least for the parameters close to the Turing domain, the system may exhibit bi-stability: the heterogeneous attractor still exists

24 L.A.D. Rodrigues et al. Fig. 20 Prey spatial distribution shown at time t = 4000 as obtained for parameter sets IV to VII, (a) to (d), respectively, and dispersal parameters (a) μ N = 0.025, μ P = 0.9, (b) μ N = 0.004, μ P = 0.95, (c) μ N = 0.02, μ P = 0.95, and (d) μ N = 0.008, μ P = 0.95 but whether the system evolves to it or return to the homogeneous equilibrium depends on the perturbation magnitude. The homogeneous equilibrium may be stable with respect to small perturbations but unstable with respect to larger ones. In order to test this hypothesis, we choose the reaction parameters as in Set I. We consider the following initial conditions. We keep the predator density at its steady state value at all sites, P x,y,0 = P for any (x, y), and we perturb the prey density at 90% of all sites: N x,y,0 = N + N for (x, y) X, N x,y,0 = N for (x, y) / X, (34) where X is the set of the perturbed lattice sites, their position being chosen randomly, and the perturbation amplitude N is the same at all perturbed sites. We run

25 Pattern Formation, Long-Term Transients Fig. 21 Critical perturbation amplitude (scaled to the equilibrium density) vs. the ratio of prey and predator dispersal rates. Solid curve for μ P = 0.7, dashed curve for μ P = 0.3 numerical simulations for different values of the perturbation amplitude. For different combinations of the dispersal parameters outside the Turing domain (cf. Fig. 4), we have discovered that, indeed, the homogeneous distribution loses its stability when the amplitude N becomes sufficiently large. The system then eventually evolves to a stationary heterogeneous spatial distribution similar to the ones shown in the bottom row of Figs. 5 and 6. To look into this matter more quantitatively, we performed simulations for different values of the dispersal coefficients ratio. For convenience, we fixed the value of μ P and varied μ N. Figure 21 shows the calculated relation between the critical perturbation amplitude 1 scaled to the prey equilibrium density, ( N/N ) cr, and the ratio μ N /μ P. Expectedly, we have found that the closer is the dispersal coefficients ratio to the domain of the Turing instability the smaller can be the perturbation amplitude sufficient to drive the system away from the homogeneous equilibrium; see the lower part of the curves. However, the fact that the curves in Fig. 21 are leaning to the right means that, for a larger μ N /μ P (which means that the system is further away from the Turing domain, cf. Fig. 4), the system stability can still be broken by a sufficiently large perturbation. 4 Concluding Remarks In this paper, we considered a predator prey system discrete in space and time. In ecological terms, it may correspond to species with distinct seasonal reproduction time (e.g. each spring) dwelling on a fragmented habitat. In order to make our model biologically more realistic, we assumed that the prey growth is hampered by the weak Allee effect. The modelling framework that we used is the Coupled Map Lattice (CML). Our main goal was to study the system dynamics, in particular scenarios of pattern formation, for the system s parameters in a vicinity of the Turing Hopf bifurcation. We first analyzed the local (non-spatial) dynamics and revealed the principal 1 Where under-critical means that the system returns to the homogeneous steady state and overcritical means that the system converges to the heterogeneous pattern.

26 L.A.D. Rodrigues et al. bifurcation structure of the system. For the spatial system (CML), we derived the conditions of the Turing instability. Equipped by these results, we then performed extensive numerical simulations. We observed a variety of different patterns, in particular, patchy patterns with different degree of spatial irregularity, labyrinthine patterns and checkboard-like two-periodic patterns. Interestingly, we observed that the interplay between the periodic or quasi-periodic local dynamics and the Turing instability results in formation of stationary patterns rather than in spatiotemporal ones. Also, we observed that, while the minus-one Turing bifurcation typically results in a spatially regular checkboard-like pattern (cf. Figs. 8 and 12), the plus-one Turing bifurcation results in irregular spatial patterns (Figs. 5, 11, and 17). Note that, since we are especially interested in the effects of the Turing bifurcation, in our study we focused on the cases when the dispersal coefficients μ N and μ P are significantly different. Indeed, it is a well-known fact that, in a two-species predator prey system, 2 the Turing instability is not possible when the diffusion or dispersal coefficients are equal. This applies both to continuous systems (Segel and Jackson 1972) and discrete systems (White and White 2005; also this paper, see Figs. 4, 10, and 15). Moreover, the Turing domain in the parameter space tends to increase with an increase in the diffusion coefficients ratio but it shrinks to a single point when this ratio approaches unity; for details, see Malchow et al. (2008). The main findings of our study are summarized below: 1. We considered the system dynamics in a vicinity of the Turing Hopf bifurcation and studied scenarios of pattern formation for the corresponding parameter values. Although the Turing Hopf bifurcation was indeed suggested as a plausible mechanism of complex spatiotemporal pattern formation in spaceand time-continuous diffusion-reaction systems both of a general type (Meixner et al. 1997; Yang and Epstein 2003; Yang et al. 2004; Alonsoetal.2007; Liu et al. 2007) and those specifically describing the predator prey dynamics (Baurmann et al. 2007), we are not aware about any study of this issue in a discrete system. We obtained that in case of the plus-one bifurcation, the emerging spatial patterns appear to be stationary in the large-time limit even when the non-spatial (single site) dynamics is oscillatory. Note that a similar behaviour can be observed in a vicinity of the Turing Hopf bifurcation in a spaceand time-continuous predator prey system as well (cf. Baurmann et al. 2007; Banerjee and Petrovskii 2010). Therefore, we conclude that this property is not a consequence of the system s discreteness but is a more general property of the spatial predator prey interactions. This is in a good agreement with recent results by Ricard and Mischler (2009) where they concluded that, in a space- and time-continuous system, the patterns tend to become stationary when the diffusion coefficient become different enough. On the contrary, the patterns emerging as a result of the minus-one bifurcation are always oscillatory and this seems to be a specific property of the discrete systems. 2 The situation is different when the number of interacting species is more than two; e.g. see Vastano et al. (1987).

27 Pattern Formation, Long-Term Transients 2. We discovered a phenomenon of long memory of the system, i.e. the existence of long-term transient that persists on the time scale of almost two orders of magnitude longer than it takes the effects of the initial conditions to disappear. 3. We showed that the pattern formation is, in fact, possible for parameters outside of the Turing domain. In that case, the spatially homogeneous population distribution is stable with respect to small heterogeneous perturbations. However, the homogeneous steady state appears to be stable only locally, but not globally. A sufficiently large perturbation will drive the system away and result in formation of a spatial pattern very similar to that observed due to the usual Turing scenario. A similar phenomenon has recently been observed in a space- and time-continuous predator prey system (Banerjee and Petrovskii 2010), which may indicate that this type of finite-amplitude instability may be rather common in spatiotemporal predator prey systems. At least some of our findings seem to have an immediate ecological interpretation. Contrary to space-time-continuous systems where non-stationary patchy patterns are a typical phenomenon (e.g. see Malchow et al. 2008), in discrete systems the spatial aspect appears to have a stabilizing impact on the system dynamics; see item 1 in the list above. It may indicate that, intrinsically, the population dynamics in a fragmented habitat tends to be stationary rather than transient and, therefore, the population fluctuations that are commonly observed in nature should be more likely attributed to the impact of external factors. On the other hand, the observed phenomenon of long-term transients means that, when a sudden change in the type of spatial pattern occurs in ecological populations dwelling in a fragmented habitat, it is not necessarily to be a result of exogenous factors as it can be just an inherent property of the population dynamics. Acknowledgement D.C.M. and L.A.D.R. were supported by grants from CAPES, process BEX 3696/09-0 and BEX 3775/09-7, respectively. References Allen, L. S. J. (2007). An introduction to mathematical biology. Upper Saddle River: Pearson Prentice Hall. Allen, J. C., Schaffer, W. M., & Rosko, D. (1993). Chaos reduces species extinction by amplifying local population noise. Nature, 364, Alonso, S., Míguez, D. G., & Sagués, F. (2007). Differential susceptibility to noise of mixed Turing and Hopf modes in a photosensitive chemical medium. Europhys. Lett., 81, 1 8. Andersen, M. (1991). Properties of some density-dependent integrodifference equation population models. Math. Biosci., 104, Banerjee, M., & Petrovskii, S. V. (2010). Self-organized spatial patterns and chaos in a ratio-dependent predator prey system. Theor. Ecol. doi: /s (in press). Baurmann, M., Gross, T., & Feudel, U. (2007). Instabilities in spatially extended predator prey systems: Spatio-temporal patterns in the neighborhood of Turing Hopf bifurcations. J. Theor. Biol., 245, Brindley, J., & Everson, R. M. (1989). Disturbance propagation in Coupled Lattice Maps. Phys. Lett. A, 134, Comins, H. N., Hassell, M. P., & May, R. M. (1992). The spatial dynamics of host-parasitoid systems. J. Anim. Ecol., 61,