Abstract
In this paper, we focus on a ratio dependent predator-prey system with self- and cross-diffusion and constant harvesting rate. By making use of a brief stability and bifurcation analysis, we derive the symbolic conditions for Hopf, Turing and wave bi- furcations of the system in a spatial domain. Additionally, we illustrate spatial pattern formations caused by these bifurcations via numerical examples. A series of numerical examples reveal that one can observe several typical spatiotemporal patterns such as spotted, spot-stripelike mixtures due to Turing bifurcation and an oscillatory wave pat- tern due to the wave bifurcation. Thus the obtained results disclose that the spatially extended system with self-and cross-diffusion and constant harvesting rate plays an important role in the spatiotemporal pattern formations in the two dimensional space.
Highlights
In population dynamics, many ecologists and mathematicians are interested in Michaelis–Menten-type predator–prey model, so-called a ratio-dependent predator–prey system [2, 6, 10, 14, 20] as follows: dU = rU dT −U K cUV mV + U dV dT V −D + f mV U +
We investigate bifurcation phenomena of a ratio dependent predator–prey system with self- and cross-diffusion and constant harvesting rate
It has been shown from the bifurcation analysis that the system has Turing, Hopf and wave bifurcations
Summary
Many ecologists and mathematicians are interested in Michaelis–. The tendency of predators would be to get closer to the prey, and the chase velocity of predators may be considered to be proportional to the dispersive velocity of the prey In this context, there has been considerable interest in investigating the stability behavior of systems of interacting population by taking into account the effect of self as well as cross-diffusion [5, 11, 17]. The predator prefers to avoid group defence by a huge number of prey and chooses to catch its prey from a smaller concentration group unable to sufficiently resist [5, 8] For this reason, it is reasonable to assume that d12 could be any number while d21 is positive. In this paper, we will focus on studying bifurcation phenomena of systems (1.5) and (1.6)
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