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We investigate spatiotemporal dynamics of a semi-ratio-dependent predator-prey system with reaction-diffusion and zero-flux boundary. We obtain the conditions for Hopf, Turing, and wave bifurcations of the system in a spatial domain by making use of the linear stability analysis and the bifurcation analysis. In addition, for an initial condition which is a small amplitude random perturbation around the steady state, we classify spatial pattern formations of the system by using numerical simulations. The results of numerical simulations unveil that there are various spatiotemporal patterns including typical Turing patterns such as spotted, spot-stripelike mixtures and stripelike patterns thanks to the Turing instability, that an oscillatory wave pattern can be emerged due to the Hopf and wave instability, and that cooperations of Turing and Hopf instabilities can cause occurrence of spiral patterns instead of typical Turing patterns. Finally, we discuss spatiotemporal dynamics of the system for several different asymmetric initial conditions via numerical simulations.

In recent years, pattern formations in nonlinear complex systems have been one of the central problems of the natural, social, technological sciences and ecological systems [

In this context, in this paper, we will focus on the following a semi-ratio-dependent predator-prey system with reaction-diffusion:

In fact, the authors in [

Throughout this paper, we assume that no external input is imposed from the outside. Hence the boundary conditions are taken as

In order to minimize the number of parameters in system (

For convenience, we set

The main object of this paper is to look into the spatial dynamic behaviors of system (

In order to investigate pattern formations of system (

a nontrivial positive stationary state

Now, to perform a linear stability analysis for the nontrivial stationary state

Hopf bifurcation is an instability induced by the transformation of the stability of a focus. In fact, the space-independent Hopf bifurcation breaks temporal symmetry of a system and gives rise to oscillations that are uniform in space and periodic in time. Mathematically speaking, the Hopf bifurcation occurs when

Turing instability (or called Turing bifurcation) is a phenomenon that causes certain reaction-diffusion system to lead to spontaneous stationary configuration. That is why Turing instability is often called

In fact, the Turing instability sets in when at least one of the solutions of (

The wave instability caused by the wave bifurcation plays an important part in pattern formations in many areas [

It is well known that, at the wave threshold

If one takes

Bifurcation diagrams of system (

Dispersion relations: (I)

In this section, we will investigate spatiotemporal pattern formations of the spatially extended system (

In order to solve partial differential equations numerically, one has to discretize the space and time of the given problem. For this reason, the discrete domain for the continuous domain

System (

It is well known that the spatiotemporal dynamics of a diffusion-reaction system depends on the choice of initial conditions [

In the numerical simulations, different types of spatiotemporal dynamics are observed and we have found that the distributions of predator and prey are always of the same type. Consequently, we can restrict our analysis of pattern formations to one distribution. In this section, we show the distribution of predator, for instance.

Now, we will classify spatiotemporal patter formations of system (

As mentioned in Section

First, we will investigate dynamical behaviors of system (

Snapshots of contour pictures of the time evolution of predator in system (

From now on, through this section, we will not display the snapshots of the initial pattern as Figure

If we take the parameter values

Snapshots of contour pictures of the time evolution of predator in system (

Snapshots of contour pictures of the time evolution of predator in system (

Next, in order to study dynamical behaviors of system (

Snapshots of contour pictures of the time evolution of predator in system (

Dynamical behavior of system (

Thus Figures

Now, consider the parameters

Snapshots of contour pictures of the time evolution of predator in system (

Dynamical behavior of system (

On the contrary, domain IV is the region of Hopf and Turing instabilities according to Figure

Snapshots of contour pictures of the time evolution of predator in system (

Snapshots of contour pictures of the time evolution of predator in system (

Various spatiotemporal patterns generated by a semi-ratio-dependent predator-prey system with reaction-diffusion are studied theoretically and numerically. The spatial parametric domain is divided into five regions by three bifurcation lines, Hopf, Turing, and wave bifurcation lines, shown in Figure

The authors in [

Firstly, we employ an initial condition that predators are introduced in a small, localized region of the circle domain while the population of prey is fixed:

Figures

Snapshots of contour pictures of the time evolution of predator in system (

Snapshots of contour pictures of the time evolution of predator in system (

These figures beg the question that the spotted pattern could be a globally stable pattern even though we do not have any theoretical evidences. Similarly, we can figure out numerically that system (

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012-0004725).