88 research outputs found
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
Analytical detection of stationary and dynamic patterns in a prey-predator model with reproductive Allee effect in prey growth
Allee effect in population dynamics has a major impact in suppressing the
paradox of enrichment through global bifurcation, and it can generate highly
complex dynamics. The influence of the reproductive Allee effect, incorporated
in the prey's growth rate of a prey-predator model with Beddington-DeAngelis
functional response, is investigated here. Preliminary local and global
bifurcations are identified of the temporal model. Existence and non-existence
of heterogeneous steady-state solutions of the spatio-temporal system are
established for suitable ranges of parameter values. The spatio-temporal model
satisfies Turing instability conditions, but numerical investigation reveals
that the heterogeneous patterns corresponding to unstable Turing eigen modes
acts as a transitory pattern. Inclusion of the reproductive Allee effect in the
prey population has a destabilising effect on the coexistence equilibrium. For
a range of parameter values, various branches of stationary solutions including
mode-dependent Turing solutions and localized pattern solutions are identified
using numerical bifurcation technique. The model is also capable to produce
some complex dynamic patterns such as travelling wave, moving pulse solution,
and spatio-temporal chaos for certain range of parameters and diffusivity along
with appropriate choice of initial conditions Judicious choices of
parametrization for the Beddington-DeAngelis functional response help us to
infer about the resulting patterns for similar prey-predator models with
Holling type-II functional response and ratio-dependent functional response
Existence of spatial patterns in reaction–diffusion systems incorporating a prey refuge
In real-world ecosystem, studies on the mechanisms of spatiotemporal pattern formation in a system of interacting populations deserve special attention for its own importance in contemporary theoretical ecology. The present investigation deals with the spatial dynamical system of a two-dimensional continuous diffusive predator–prey model involving the influence of intra-species competition among predators with the incorporation of a constant proportion of prey refuge. The linear stability analysis has been carried out and the appropriate condition of Turing instability around the unique positive interior equilibrium point of the present model system has been determined. Furthermore, the existence of the various spatial patterns through diffusion-driven instability and the Turing space in the spatial domain have been explored thoroughly. The results of numerical simulations reveal the dynamics of population density variation in the formation of isolated groups, following spotted or stripe-like patterns or coexistence of both the patterns. The results of the present investigation also point out that the prey refuge does have significant influence on the pattern formation of the interacting populations of the model under consideration
Cross-diffusion induced Turing instability for a three species food chain model
AbstractIn this paper, we study a strongly coupled reaction–diffusion system describing three interacting species in a food chain model, where the third species preys on the second one and simultaneously the second species preys on the first one. We first show that the unique positive equilibrium solution is globally asymptotically stable for the corresponding ODE system. The positive equilibrium solution remains linearly stable for the reaction–diffusion system without cross-diffusion, hence it does not belong to the classical Turing instability scheme. We further proved that the positive equilibrium solution is globally asymptotically stable for the reaction–diffusion system without cross-diffusion by constructing a Lyapunov function. But it becomes linearly unstable only when cross-diffusion also plays a role in the reaction–diffusion system, hence the instability is driven solely from the effect of cross-diffusion. Our results also exhibit some interesting combining effects of cross-diffusion, intra-species competitions and inter-species interactions
The diffusion-driven instability and complexity for a single-handed discrete Fisher equation
For a reaction diffusion system, it is well known that the diffusion coefficient of the
inhibitor must be bigger than that of the activator when the Turing
instability is considered. However, the diffusion-driven instability/Turing
instability for a single-handed discrete Fisher equation with the Neumann
boundary conditions may occur and a series of 2-periodic patterns have been
observed. Motivated by these pattern formations, the existence of 2-periodic
solutions is established. Naturally, the periodic double and the chaos
phenomenon should be considered. To this end, a simplest two elements system
will be further discussed, the flip bifurcation theorem will be obtained by
computing the center manifold, and the bifurcation diagrams will be
simulated by using the shooting method. It proves that the Turing
instability and the complexity of dynamical behaviors can be completely
driven by the diffusion term. Additionally, those effective methods of
numerical simulations are valid for experiments of other patterns, thus, are
also beneficial for some application scientists
Nonconstant positive steady states and pattern formation of a diffusive epidemic model
It is our purpose in this paper to make a detailed description for the structure of the set of the nonconstant steady states for the two-dimensional epidemic S-I model with diffusion incorporating demographic and epidemiological processes with zeroflux boundary conditions. We first study the conditions of diffusion-driven instability occurrence, which induces spatial inhomogeneous patterns. The results will extend to the derivative of prey’s functional response with prey is positive. Moreover, we establish the local and global structure of nonconstant positive steady state solutions. A priori estimates for steady state solutions will play a key role in the proof. Our results indicate that the diffusion has a great influence on the spread of the epidemic and extend well the finding of spatiotemporal dynamics in the epidemic model
Pattern Formation in a Bacterial Colony Model
We investigate the spatiotemporal dynamics of a bacterial colony model. Based on the stability analysis, we derive the conditions for Hopf and Turing bifurcations. Furthermore, we present novel numerical evidence of time evolution of patterns controlled by parameters in the model and find that the model dynamics exhibit a diffusion controlled formation growth to spots, holes and stripes pattern replication, which show that the bacterial colony model is useful in revealing the spatial predation dynamics in the real world
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