144 research outputs found
Temporal and spatial patterns in a diffusive ratio-dependent predator–prey system with linear stocking rate of prey species
The ratio-dependent predator–prey model exhibits rich interesting dynamics due to the singularity of the origin. It is one of prototypical pattern formation models. Stocking in a ratio-dependent predator–prey models is relatively an important research subject from both ecological and mathematical points of view. In this paper, we study the temporal, spatial patterns of a ratio-dependent predator–prey diffusive model with linear stocking rate of prey species. For the spatially homogeneous model, we derive conditions for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solution by the center manifold and the normal form theory. For the reaction-diffusion model, firstly it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Then it is demonstrated that the model exhibits Hopf bifurcation which produces temporal inhomogeneous patterns. Finally, the non-existence and existence of positive non-constant steady-state solutions are established. We can see spatial inhomogeneous patterns via Turing instability, temporal periodic patterns via Hopf bifurcation and spatial patterns via the existence of positive non-constant steady state. Moreover, numerical simulations are performed to visualize the complex dynamic behavior
Temporal and spatial patterns in a diffusive ratio-dependent predator-prey system with linear stocking rate of prey species
The ratio-dependent predator–prey model exhibits rich interesting dynamics due to the singularity of the origin. It is one of prototypical pattern formation models. Stocking in ratio-dependent predator–prey models is relatively an important research subject from both ecological and mathematical points of view. In this paper, we study the temporal, spatial patterns of a ratio-dependent predator–prey diffusive model with linear stocking rate of prey species. For the spatially homogeneous model, we derive conditions for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solution by the center manifold and the normal form theory. For the reaction-diffusion model, firstly it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Then it is demonstrated that the model exhibits Hopf bifurcation which produces temporal inhomogeneous patterns. Finally, the non-existence and existence of positive non-constant steady-state solutions are established. We can see spatial inhomogeneous patterns via Turing instability, temporal periodic patterns via Hopf bifurcation and spatial patterns via the existence of positive non-constant steady state. Moreover, numerical simulations are performed to visualize the complex dynamic behavior
The influence of dispersal on a predator-prey system with two habitats
Dispersal between different habitats influences the dynamics and stability of
populations considerably. Furthermore, these effects depend on the local
interactions of a population with other species. Here, we perform a general and
comprehensive study of the simplest possible system that includes dispersal and
local interactions, namely a 2-patch 2-species system. We evaluate the impact
of dispersal on stability and on the occurrence of bifurcations, including
pattern forming bifurcations that lead to spatial heterogeneity, in 19
different classes of models with the help of the generalized modelling
approach. We find that dispersal often destabilizes equilibria, but it can
stabilize them if it increases population losses. If dispersal is nonrandom,
i.e. if emigration or immigration rates depend on population densities, the
correlation of stability with migration rates is positive in part of the
models. We also find that many systems show all four types of bifurcations and
that antisynchronous oscillations occur mostly with nonrandom dispersal
Diffusion-driven instabilities and emerging spatial patterns in patchy landscapes
Spatial variation in population densities across a landscape is a feature of many ecological systems, from
self-organised patterns on mussel beds to spatially restricted insect outbreaks. It occurs as a result of
environmental variation in abiotic factors and/or biotic factors structuring the spatial distribution of
populations. However the ways in which abiotic and biotic factors interact to determine the existence
and nature of spatial patterns in population density remain poorly understood. Here we present a new
approach to studying this question by analysing a predator–prey patch-model in a heterogenous
landscape. We use analytical and numerical methods originally developed for studying nearest-
neighbour (juxtacrine) signalling in epithelia to explore whether and under which conditions patterns
emerge. We find that abiotic and biotic factors interact to promote pattern formation. In fact, we find a
rich and highly complex array of coexisting stable patterns, located within an enormous number of
unstable patterns. Our simulation results indicate that many of the stable patterns have appreciable
basins of attraction, making them significant in applications. We are able to identify mechanisms for
these patterns based on the classical ideas of long-range inhibition and short-range activation, whereby
landscape heterogeneity can modulate the spatial scales at which these processes operate to structure
the populations
Dynamic analysis of a Leslie–Gower-type predator–prey system with the fear effect and ratio-dependent Holling III functional response
In this paper, we extend a Leslie–Gower-type predator–prey system with ratio-dependent Holling III functional response considering the cost of antipredator defence due to fear. We study the impact of the fear effect on the model, and we find that many interesting dynamical properties of the model can occur when the fear effect is present. Firstly, the relationship between the fear coefficient K and the positive equilibrium point is introduced. Meanwhile, the existence of the Turing instability, the Hopf bifurcation, and the Turing–Hopf bifurcation are analyzed by some key bifurcation parameters. Next, a normal form for the Turing–Hopf bifurcation is calculated. Finally, numerical simulations are carried out to corroborate our theoretical results
Turing instability in a diffusive predator-prey model with multiple Allee effect and herd behavior
Diffusion-driven instability and bifurcation analysis are studied in a
predator-prey model with herd behavior and quadratic mortality by incorporating
multiple Allee effect into prey species. The existence and stability of the
equilibria of the system are studied. And bifurcation behaviors of the system
without diffusion are shown. The sufficient and necessary conditions for Turing
instability occurring are obtained. And the stability and the direction of Hopf
and steady state bifurcations are explored by using the normal form method.
Furthermore, some numerical simulations are presented to support our
theoretical analysis. We found that too large diffusion rate of prey prevents
Turing instability from emerging. Finally, we summarize our findings in the
conclusion
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