Wave train selection behind invasion fronts in reaction–diffusion predator–prey models
Introduction
The cause of temporal cycles in natural populations has been a focus of study by ecologists for many decades. A classical hypothesis is that this oscillatory behaviour arises from the interaction between a predator population and its prey, and many models have been constructed and studied to support this hypothesis (see, for example [1]). Such models have often taken the form of kinetics systems: ordinary differential equation models that describe the time evolution of predator and prey densities that are assumed to be spatially constant. More recently, however, field studies have shown that in some natural populations oscillations are not synchronized in space, and when viewed in one spatial dimension take the form of a wave train [2], [3], [4], [5], [6], [7]. Wave trains, or periodic travelling waves, are spatio-temporal patterns that are periodic in both time and space and have the appearance of a spatially periodic solution that maintains its shape and moves at a constant speed. Consequently, there has been a great deal of study recently on oscillatory reaction–diffusion systems because these partial differential equation models possess wave train solutions (see [8] and references therein).
One way that wave trains can arise in oscillatory reaction–diffusion systems is following a predator invasion [9], [10], [11]. The initial condition for such a scenario consists of the prey at carrying capacity everywhere in the spatial domain, except in a localized region in which a predator is introduced. Typically, a travelling front evolves that maintains its shape and moves at a constant speed. In some cases, behind this primary invasion front a secondary transition occurs, and the solution takes the form of a wave train. Two numerical simulations where wave trains evolve following a predator invasion are illustrated in Fig. 1. We can see from these examples that the wave train behind the front does not necessarily move at the same speed, or even in the same direction, as the invasion front itself.
For oscillatory reaction–diffusion systems near a Hopf bifurcation in the corresponding kinetics, there exists a one-parameter family of wave train solutions and a range of corresponding speeds [12]. In a particular numerical simulation of an invasion, we typically observe only a single member of this family, and this seems robust to changes in initial or boundary conditions. Therefore, it appears that a particular wave train is somehow selected out of the family. We would like to find some means of predicting the selected wave train.
Sherratt has in fact already produced an explanation of the selection mechanism and a prediction for the wave train selected behind invasion fronts in reaction–diffusion systems with oscillatory kinetics [13]. The basis of his prediction is an approximating lambda–omega () system. The behaviour of an oscillatory reaction–diffusion system near a nondegenerate supercritical Hopf bifurcation can be described by the simpler system whose coefficients are obtained from the normal form of the Hopf bifurcation in the kinetics system. Predictions derived in this way are applicable near the Hopf bifurcation and when the predator and prey have nearly equal diffusion coefficients. For more widely applicable predictions, such as in cases where there are larger amplitude oscillations or unequal diffusion coefficients, it would be beneficial to develop a criterion to predict the selected wave train that does not directly depend on the system.
In the remainder of this paper, we derive and test such a criterion. We first introduce in Section 2 the class of two-component reaction–diffusion systems we consider. These systems describe the evolution of population density distributions of two species, one a prey and the other a predator, in one space dimension. Two spatially homogeneous steady states are relevant: an unstable prey-only state that is invaded by a travelling front, and a coexistence state unstable to oscillatory modes that interacts with the invasion. In some cases, such as illustrated in Fig. 1(a), there is a secondary front that invades the coexistence state. The speed of a front invading an unstable steady state can be predicted by the linear spreading speed (see the review [14] and references therein) which depends only on linearization about the unstable state. In Section 3, we consider coherent structures in the complex Ginzburg–Landau (CGL) equation [14], [15], [16], [17], [18], of which the system is a special case. The unstable state in this case is the origin, which corresponds to the coexistence state in predator–prey systems, and coherent structures represent travelling fronts that connect the steady state to wave trains. The linear spreading speed selects a particular coherent structure and wave train, and this retrieves the prediction developed in [13]. Coherent structures have been generalized as defects in general reaction–diffusion systems by Sandstede and Scheel in [19]. In Section 4, we extend the prediction for the system to a new “pacemaker” criterion for defects in predator–prey reaction–diffusion systems that connect the unstable prey-only state with wave trains associated with oscillatory instabilities of the coexistence state. For the speed of the selected defect we take the minimum of the linear spreading speeds for the prey-only and coexistence states, and for the frequency of the selected wave train measured in the frame comoving with the defect we take the frequency of the linear Hopf instability of the coexistence state. The performance of the pacemaker criterion is then numerically tested in Section 5 on sample oscillatory reaction–diffusion systems. We find that the pacemaker criterion gives accurate predictions for a wider range of parameter values than the criterion does, but still falls off in accuracy farther away from the Hopf bifurcation. Finally, Section 6 discusses and summarizes the key results.
Section snippets
Mathematical background
We consider predator–prey reaction–diffusion systems in one space dimension, of the form where is the density of prey at position and time and is the density of predator at . Both and are real-valued functions. The positive parameters and are the diffusion coefficients of the prey and predator, respectively, while the functions and depend on parameters not explicitly shown here, and describe the local
Coherent structures and selection in the lambda–omega system
In this section, we use the linear spreading speed of a coherent structure in the system to find the wave train selected behind an invading front, and thus recover the prediction (8) of [13]. This derivation is implicit in the physics literature cited, but we reproduce it here to motivate the more general selection criterion we use in the case when the system is not a good approximation to (1). This derivation also points out that the selected wave train only requires that initial
Beyond the lambda–omega system: the pacemaker criterion
In this section, we consider wave train selection behind invading fronts in the predator–prey reaction–diffusion systems described in Section 2. In general, for the full oscillatory reaction–diffusion systems (1), wave trains are not sinusoidal and we do not have exact solutions for them, so a prediction of the form (8) is not possible. However, as shown in Fig. 1, numerical simulations of predator invasions in the full system even well away from the Hopf bifurcation in the kinetics still
The criterion in practice
To study the validity of criterion (16), we have considered three particular forms for the kinetic equations in the oscillatory reaction–diffusion system (1) as well as a system. That is, we consider the four sets of kinetics where , , , , , and are parameters. Models (ii)–(iv) all
Discussion
The pacemaker criterion (16) developed here provides a relatively simple method for predicting the selection of wave trains following predator invasions in oscillatory reaction–diffusion models. Our study of the performance of the pacemaker criterion for models (ii)–(iv) suggests that this criterion may in general give a good prediction for some range of parameters near the Hopf bifurcation in the kinetics. However, the performance of the pacemaker criterion is clearly model dependent, and
Acknowledgements
This work was partially supported by funding from the Natural Sciences and Engineering Research Council of Canada (NSERC). In addition, S.M. Merchant would like to acknowledge the Pacific Institute for the Mathematical Sciences (PIMS) for funding through the IGTC fellowship program. We would also like to thank Michael Doebeli and his laboratory group for many helpful comments and suggestions. Finally, we are grateful for the comments and suggestions of two anonymous referees as we think they
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