Stability and Hopf bifurcation for a delayed cooperation diffusion system with Dirichlet boundary conditions

Li, Wan–Tong; Yan, Xiang-Ping; Zhang, Cun-Hua

doi:10.1016/j.chaos.2006.11.015

Cited by 34 publications

(23 citation statements)

References 8 publications

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“…Moreover, our results are comparable to those obtained in [8] in which the authors solved problem (1.5)-(1.6) using the method of steps (where one transforms the DPDE into a system of ordinary PDEs) and…”

Section: Discussion and Future Planssupporting

confidence: 84%

“…Then when τ passes τ ⋆ , the real parts of these eigenvalues pass to the positive real axis causing the Hopf bifurcating solution to be unstable. We summarize these features of the solution via the existence and stability of a positive equilibrium following the works in [17,8,1].…”

Section: Existence and Stability Of Equilibriamentioning

confidence: 99%

“…Under the assumption that the two species have the same diffusivity (i.e., λ 1 = λ 2 = λ) and same growth rates (i.e., r 1 = r 2 = r), Li et al [8] used a transformation of variables to reduce the system (1.1)-(1.2) to the form…”

Section: Introductionmentioning

confidence: 99%

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A fitted numerical method for a system of partial delay differential equations

Bashier

Patidar

2011

Computers & Mathematics with Applications

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System of partial delay differential equations Ecological models Bifurcation analysis Method of lines Fitted operator finite difference method Convergence analysis a b s t r a c t We consider a system of two coupled partial delay differential equations (PDDEs) describing the dynamics of two cooperative species. The original system is reduced to a system of ordinary delay differential equations (DDEs) obtained by applying the method of lines. Then we construct a fitted operator finite difference method (FOFDM) to solve this resulting system. The model considered in this paper is very sensitive to small changes in the parameters associated in with the model. Depending on the values of these parameters, the solution can be stable, periodic and/or aperiodic. Such behavior of the solution is exploited via the proposed FOFDM. This FOFDM is analyzed for convergence and it is seen that this method is unconditionally stable and has the accuracy of O(k + h 2 ), where k and h denote time and space step-sizes, respectively. Some numerical results confirming theoretical observations are also presented. These results are comparable with those obtained in the literature.

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Section: Discussion and Future Planssupporting

confidence: 84%

Section: Existence and Stability Of Equilibriamentioning

confidence: 99%

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A fitted numerical method for a system of partial delay differential equations

Bashier

Patidar

2011

Computers & Mathematics with Applications

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“…Delays may also occur as a consequence of developmental time and/or interaction between individuals of di erent stages (Royama 1981, Hastings 1984. In mathematical terms, the destabilisation of a positive steady state, both in nonspatial and spatial systems, usually occurs through the Hopf bifurcation (Green and Stech 1981, Fowler 1982, Busenberg and Huang 1996, Li et al 2008, Su et al 2009) that leads to limit-cycle oscillatory behaviour. We menton here that such destabilization does not always happen; in particular, if the population growth is damped by a strong Allee e ect, an increase in time delay does not necessarily lead to the Hopf bifurcation, e.g.…”

Section: Introductionmentioning

confidence: 99%

Delay driven spatiotemporal chaos in single species population dynamics models

Jankovic

Petrovskii

Banerjee

2016

Theoretical Population Biology

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Questions surrounding the prevalence of complex population dynamics form one of the central themes in ecology. Limit cycles and spatiotemporal chaos are examples that have been widely recognised theoretically, although their importance and applicability to natural populations remains debatable. The ecological processes underlying such dynamics are thought to be numerous, though there seems to be consent as to delayed density dependence being one of the main driving forces. Indeed, time delay is a common feature of many ecological systems and can signi cantly in uence population dynamics. In general, time delays may arise from interand intra-speci c trophic interactions or population structure, however in the context of single species populations they are linked to more intrinstic biological phenomena such as gestation or resource regeneration. In this paper, we consider theoretically the spatiotemporal dynamics of a single species population using two di erent mathematical formulations. Firstly, we revisit the di usive logistic equation in which the per capita growth is a function of some speci ed delayed argument. We then modify the model by incorporating a spatial convolution which results in a biologically more viable integro-di erential model. Using the combination of analytical and numerical techniques, we investigate the e ect of time delay on pattern formation. In particular, we show that for su ciently large values of time delay the system's dynamics are indicative to spatiotemporal chaos. The chaotic dynamics arising in the wake of a travelling population front can be preceded by either a plateau corresponding to dynamical stabilisation of the unstable equilibrium or by periodic oscillations.

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“…Hence, under the Dirichlet boundary conditions, it is very difficult to study the stability of nonconstant steady-state solutions because in this case the analysis of the characteristic equation is very difficult [1,8,16,19]. The main goal of this paper is to study the stability of the bifurcating positive steady-state solutions of system (1.1).…”

Section: Introductionmentioning

confidence: 99%

Stability of Positive Steady-State Solutions in a Delayed Lotka-Volterra Diffusion System

Yan¹,

Zhang²

2012

Journal of the Korean Mathematical Society

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Abstract. This paper considers the stability of positive steady-state solutions bifurcating from the trivial solution in a delayed Lotka-Volterra two-species predator-prey diffusion system with a discrete delay and subject to the homogeneous Dirichlet boundary conditions on a general bounded open spatial domain with smooth boundary. The existence, uniqueness and asymptotic expressions of small positive steady-sate solutions bifurcating from the trivial solution are given by using the implicit function theorem. By regarding the time delay as the bifurcation parameter and analyzing in detail the eigenvalue problems of system at the positive steady-state solutions, the asymptotic stability of bifurcating steady-state solutions is studied. It is demonstrated that the bifurcating steady-state solutions are asymptotically stable when the delay is less than a certain critical value and is unstable when the delay is greater than this critical value and the system under consideration can undergo a Hopf bifurcation at the bifurcating steady-state solutions when the delay crosses through a sequence of critical values.

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Stability and Hopf bifurcation for a delayed cooperation diffusion system with Dirichlet boundary conditions

Cited by 34 publications

References 8 publications

A fitted numerical method for a system of partial delay differential equations

A fitted numerical method for a system of partial delay differential equations

Delay driven spatiotemporal chaos in single species population dynamics models

Stability of Positive Steady-State Solutions in a Delayed Lotka-Volterra Diffusion System

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