Abstract
We investigate a spatial ratio-dependent predator-prey system with linear harvesting rate. By using linear stability and bifurcation analysis, we obtain the conditions for Hopf and Turing bifurcation in the spatial domain. In addition, we classify spatial pattern formations of the system by making use of numerical simulations. In fact, the numerical simulations unveil that the typical Turing patterns, such as spotted, spot-stripelike mixtures and stripelike patterns, can be observed even if the system has the linear harvesting rate. In order to analyze these patterns via the spatial frequency, the discrete Fourier transform is used. Moreover, we discuss that the linear harvesting system is more realistic than a predator-prey system with constant harvesting. Our results disclose that the spatially extended system with linear harvesting rate has more complex dynamic patterns in the two-dimensional space. It may help to understand the effects of harvesting on species in the real world.
Highlights
Since a classical Holling-Tanner type predator-prey system, so-called a ratio-dependent predator-prey system, has received great attention among mathematicians and mathematical biologists, we will focus our concern on the following system of nonlinear coupled ordinary differential equations: dN dt = rN (1 − N K ) cPN N + mP dP dt −δP + sPN N + mP (1)where N and P stand for prey and predator, respectively
We adopt a ratio-dependent functional response in system (1) which roughly the per capita predator growth rate should be a function of the ratio of prey to predator abundant
We have investigated pattern formations of a ratio-dependent predator-prey system with linear harvesting rate
Summary
Since a classical Holling-Tanner type predator-prey system, so-called a ratio-dependent predator-prey system, has received great attention among mathematicians and mathematical biologists (see [1,2,3,4]), we will focus our concern on the following system of nonlinear coupled ordinary differential equations: dN dt. In this context, recently, many authors in [9,10,11,12,13,14,15] have investigated spatiotemporal pattern formations and bifurcation analysis of predator-prey systems with reaction diffusion. Many researchers have studied complex bifurcation phenomena and dynamics of the models with nonzero constant rate of harvesting of either species or both species simultaneously (see [7, 8, 16,17,18]). It is needed to take into account nonconstant harvesting rate Such harvesting activity has an effect on both species, prey and predator, directly. With the ideas discussed above, in the paper, we consider the following ratiodependent predator-prey system with reaction-diffusion and linear harvesting rates h and u: aPN N+P hN,. We discuss that the linear harvesting rate is more realistic than the constant harvesting by comparing the above linear harvesting system with the constant harvesting system studied by Zhang et al in [15]
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