Stable Limit Cycles in Prey-Predator Populations

Albrecht, Felix; Gatzke, Harry; Wax, Nelson; May, Robert M.

doi:10.1126/science.181.4104.1073

Cited by 42 publications

(23 citation statements)

References 2 publications

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“…In [40], Rosenzweig argued that enrichment of the environment (larger carrying capacity k in (1.3)) leads to destabilizing of the coexistence equilibrium, which is the so-called paradox of enrichment. May [34] pointed out the importance of the limit cycle in the population dynamics, but the uniqueness of the limit cycle turns out to be a difficult mathematical question (see Albrecht et al [1]). Hsu, Hubbell and Waltman [20,23] considered the global stability of coexistence equilibrium and Cheng [3] first proved the uniqueness of limit cycle of (1.3) (see also [28,50]).…”

mentioning

confidence: 99%

Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator–prey system

Wei

Shi

2009

Journal of Differential Equations

470

330

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A diffusive predator-prey system with Holling type-II predator functional response subject to Neumann boundary conditions is considered. Hopf and steady state bifurcation analysis are carried out in details. In particular we show the existence of multiple spatially non-homogeneous periodic orbits while the system parameters are all spatially homogeneous. Our results and global bifurcation theory also suggest the existence of loops of spatially non-homogeneous periodic orbits and steady state solutions. These results provide theoretical evidences to the complex spatiotemporal dynamics found by numerical simulation.

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confidence: 99%

Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator–prey system

Wei

Shi

2009

Journal of Differential Equations

470

330

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“…The existence and number of limit cycles are important topics in the study of most applied mathematical models. Such study has made possible a better understanding of many real world oscillatory phenomena in nature [1,11,17]. For predator-prey systems, it is well known that the existence of limit cycles is related to the existence, stability, and bifurcation of a positive equilibrium.…”

mentioning

confidence: 99%

Multiple Focus and Hopf Bifurcations in a Predator-Prey System with Nonmonotonic Functional Response

Xiao¹,

Zhu²

2006

SIAM J. Appl. Math.

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Abstract. In this paper, we develop a criterion to calculate the multiplicity of a multiple focus for general predator-prey systems. As applications of this criterion, we calculate the largest multiplicity of a multiple focus in a predator-prey system with nonmonotonic functional response p(x) = x ax 2 +bx+1 studied by Zhu, Campbell, and Wolkowicz [SIAM J. Appl. Math., 63 (2002), pp. 636-682] and prove that the degenerate Hopf bifurcation is of codimension two. Furthermore, we show that there exist parameter values for which this system has a unique positive hyperbolic stable equilibrium and exactly two limit cycles, the inner one unstable and outer one stable. Numerical simulations for the existence of the two limit cycles bifurcated from the multiple focus are also given in support of the criterion.Key words. predator-prey, Liénard system, multiple focus, Hopf bifurcation, codimension two, limit cycles AMS subject classifications. Primary, 34C25, 92D25; Secondary, 58F14 DOI. 10.1137/0506234491. Introduction. The existence and number of limit cycles are important topics in the study of most applied mathematical models. Such study has made possible a better understanding of many real world oscillatory phenomena in nature [1,11,17]. For predator-prey systems, it is well known that the existence of limit cycles is related to the existence, stability, and bifurcation of a positive equilibrium. In a positively invariant region, if there exists a unique positive equilibrium which is unstable, then there must exist at least one limit cycle according to the theory of Poincaré-Bendixson. On the other hand, if the unique positive equilibrium of a predator-prey system is locally stable but not hyperbolic, there might be more than one limit cycle created via Hopf bifurcation(s). Numerical simulations of Hofbauer and So [8] indicated that this is indeed the case: there can exist at least two limit cycles for some two-dimensional predator-prey systems. It was proved by Zhu, Campbell, and Wolkowicz [26] that a predator-prey system with nonmonotonic functional response can undergo a degenerate Hopf bifurcation which produces two limit cycles, and the system can also have two limit cycles through a saddle node bifurcation of limit cycles. These two limit cycles can disappear through either supercritical Hopf or/and homoclinic bifurcations. Kuang [9] and Wrzosek [22] also observed that some predator-prey systems can even have more than two limit cycles. We shall point out that in all the models studied in [8,9,22] the death rate of the predator is nonlinear, but ours in [19,26] is linear.Hopf bifurcation theory is a powerful tool for studying the existence, number, and properties of the limit cycles in mathematical biology. However, the largest number

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“…Besides, as pointed out by Keller and Segel (1970), whenever possible and certainly as a first attempt, there is always some merit in adopting the simplest reasonable model to describe a specific phenomenon. We further observe that our results of the previous section show May's robustness assertion to be wrong in regard to the global stability property of the locally stable equilibrium point, while Albrecht et al (1973) showed it to be wrong with respect to the globally stable limit cycle property; May (1973b) tended to regard their admonishments as being merely pedagogical and unlikely to be of biological interest, since he concluded that one can always construct pathological neutrally stable periodic solutions, usually by choosing growth rates whose density dependence exhibits discontinuities in its higher derivatives (May and Leonard, 1975), and that such structurally unstable models have little biological significance. Given that mean temperature is a biologically meaningful parameter for arthropod predator/prey systems in seasonal climates, and outbreaks do occur naturally in such communities, it would obviously be a little more difficult to discredit our counter-example simply by citing a lack of biological significance.…”

Section: Discussion Of Outbreak Models and Comparison Of Theory With mentioning

confidence: 74%

“…Since the various biological and mathematical interpretations of such Kolmogorov-type systems will play a fundamental role in the discussion of our results, we now briefly describe what is meant by our system of (2.2) being of this type. In particular, our specific exploitation system is of the Kolmogorov-type because the functions F and G defined in (2.2c) satisfy the following conditions (Albrecht et al, 1973;May, 1973a,b;Rescigno and Richardson, 1973 ): Note that these two representations are virtually coincident for the exact temperature range over which the linear interpolation of (2.6) may be presumed to be valid.…”

Section: Nondimensional Temperature-dependent Holling-tanner Mite Modelmentioning

confidence: 98%

Temperature-mediated stability of the interaction between spider mites and predatory mites in orchards

1988

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The nonlinear behavior of the Holling-Tanner predatory-prey differential equation system, employed by R.M. May to illustrate the apparent robustness of Kolmogorov's Theorem when applied to such exploitation systems, is re-examined by means of the numerical bifurcation code AUTO 86 with model parameters chosen appropriately for a temperature-dependent mite interaction on fruit trees. The most significant result of this analysis is that there exists a temperature range wherein multiple stable states can occur, in direct violation of May's interpretation of this system's satisfaction of Kolmogorov's Theorem: namely, that linear stability predictions have global consequences. In particular these stable states consist of a focus (spiral point) and a limit cycle separated from each other in the phase plane by an unstable limit cycle, all of which are associated with the single community equilibrium point of the system. The ecological implications of such metastability, hysteresis, and threshold behavior for the occurrence of outbreaks, the persistence of oscillations, the resiliency of the system, and the biological control of mite populations are discussed.

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Stable Limit Cycles in Prey-Predator Populations

Cited by 42 publications

References 2 publications

Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator–prey system

Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator–prey system

Multiple Focus and Hopf Bifurcations in a Predator-Prey System with Nonmonotonic Functional Response

Temperature-mediated stability of the interaction between spider mites and predatory mites in orchards

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