Stability and Oscillations in Delay Differential Equations of Population Dynamics

Gopalsamy, K.

doi:10.1007/978-94-015-7920-9

Cited by 1,643 publications

(1,067 citation statements)

References 144 publications

(206 reference statements)

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“…[7,17,29,30]) is a fundamental population model. More realistic predator-prey systems were introduced by Holling [11] using the three kinds of functional responses for different species to model the phenomena of predation.…”

Section: Introductionmentioning

confidence: 99%

Resonance and bifurcation in a discrete-time predator–prey system with Holling functional response

Ghaziani

Govaerts

Sonck

2012

Nonlinear Analysis: Real World Applications

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Abstract. We perform a bifurcation analysis of a discrete predator-prey model with Holling functional response. We summarize stability conditions for the three kinds of fixed points of the map, further called F1, F2 and F3 and collect complete information on this in a single scheme. In the case of F2 we also compute the critical normal form coefficient of the flip bifurcation analytically. We further obtain new information about bifurcations of the cycles with periods 2, 3, 4, 5, 8 and 16 of the system by numerical computation of the corresponding curves of fixed points and codim-1 bifurcations, using the software package MatContM. Numerical computation of the critical normal form coefficients of the codim-2 bifurcations enables us to determine numerically the bifurcation scenario around these points as well as possible branch switching to curves of codim-1 points. Using parameter-dependent normal forms, we compute codim-1 bifurcation curves that emanate at codim-2 bifurcation points in order to compute the stability boundaries of cycles with periods 4, 5, 8 and 16.

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Section: Introductionmentioning

confidence: 99%

Resonance and bifurcation in a discrete-time predator–prey system with Holling functional response

Ghaziani

Govaerts

Sonck

2012

Nonlinear Analysis: Real World Applications

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“…Strong interest in (1) and (1) is motivated by the fact that they represent the discrete analogues of the delay differential equations (see [22,37,38] and the references cited therein):…”

Section: Introductionmentioning

confidence: 99%

Oscillation Conditions for Difference Equations with a Monotone or Nonmonotone Argument

Moremedi

Stavroulakis

2018

Discrete Dynamics in Nature and Society

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Consider the first-order delay difference equation with a constant argument Δ ( ) + ( ) ( − ) = 0, = 0, 1, 2, . . . , and the delay difference equation with a variable argument Δ ( ) + ( ) ( ( )) = 0, = 0, 1, 2, . . . , where ( ) is a sequence of nonnegative real numbers, is a positive integer, Δ ( ) = ( + 1) − ( ), and ( ) is a sequence of integers such that ( ) ≤ − 1 for all ≥ 0 and lim →∞ ( ) = ∞. A survey on the oscillation of all solutions to these equations is presented. Examples illustrating the results are given.

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“…Stability analysis of dynamical systems using differential and difference inequalities attracted a prominent attention in the existing literature (see [1]- [12] and references therein). For stability analysis of the delay differential equation…”

Section: Introductionmentioning

confidence: 99%

Halanay type inequalities on time scales with applications

Adıvar¹,

Bohner²

2011

Nonlinear Analysis: Theory, Methods & Applications

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This paper aims to introduce Halanay type inequalities on time scales. By means of these inequalities we derive new global stability conditions for nonlinear dynamic equations on time scales. Giving several examples we show that beside generalization and extension to q-difference case, our results also provide improvements for the existing theory regarding differential and difference inequalites, which are the most important particular cases of dynamic inequalities on time scales.

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Stability and Oscillations in Delay Differential Equations of Population Dynamics

Cited by 1,643 publications

References 144 publications

Resonance and bifurcation in a discrete-time predator–prey system with Holling functional response

Resonance and bifurcation in a discrete-time predator–prey system with Holling functional response

Oscillation Conditions for Difference Equations with a Monotone or Nonmonotone Argument

Halanay type inequalities on time scales with applications

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