Prediction of predator–prey populations modelled by perturbed ODEs

Froda, Sorana; Nkurunziza, Sévérien

doi:10.1007/s00285-006-0051-9

Cited by 12 publications

(5 citation statements)

References 17 publications

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“…Also, it seems important to consider more sophisticated stochastic models, for example, some form of mild dependence as in Froda and Nkurunziza (2007), and develop appropriate inference procedures. For now, we feel that the present simple idea could prove to be a useful preliminary tool in assessing the nature of the predator-prey interaction.…”

Section: Discussionmentioning

confidence: 99%

“…Moreover, we can suppose that, for t = t , the couples (U t , V t ) and (U t , V t ) are either independent identically distributed (i.i.d.) random variables or with a known dependence structure (see, e.g., the error model proposed in Froda and Nkurunziza (2007). In what follows, we consider only the i.i.d.…”

Section: Three Classical Deterministic Models and A Stochastic Versionmentioning

confidence: 99%

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Simple testing procedures for the Holling type II model

Froda

Zahedi

2009

Theor Ecol

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In this paper, we consider predator-prey data that can be viewed as solutions to a planar system of ordinary differential equations (ODE) observed with random error. The ODE system admits a limit cycle, while the random error is supposed to act additively in the log-scale. One of the oldest such systems is Holling's type II model. In spite of its simplicity, it is still very popular in data analyses, although more sophisticated models have been introduced in the literature. We propose a simple way of deciding whether a set of predator-prey pairs is indicative or not of a departure from this basic model by exploiting the geometric properties of the solution in the phase plane. To illustrate our method, we use simulated and real data.

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Section: Discussionmentioning

confidence: 99%

Section: Three Classical Deterministic Models and A Stochastic Versionmentioning

confidence: 99%

Simple testing procedures for the Holling type II model

Froda

Zahedi

2009

Theor Ecol

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“…By using the first part of observations (44 and 42 pairs of observations for the first and second data sets respectively), we apply the estimation method presented in Froda and Nkurunziza [24]. We get estimate of (γ, β, δ, α) and (σ, φ) as given in Table 1.…”

Section: Test On Real Data Sets (Mink-muskrat and Paramecium-didinium)mentioning

confidence: 99%

Likelihood ratio test for a special predator-prey system

Nkurunziza

2008

Statistics

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In this paper, we study an inference problem for a stochastic model where the deterministic LotkaVolterra system of ordinary differential equations (ODE) is perturbed with random error. The deterministic system describes the ecological interaction between prey and predator, but depends on unknown parameters. More precisely, we consider testing problems concerning the interaction parameters of the ODE. By assuming that the random errors follow correlated Ornstein-Uhlenbeck processes, we propose a likelihood ratio test and study the asymptotic properties of this test. Finally, we perform some simulation studies that corroborate our theoretical results and we apply the suggested test to two real data sets (Canadian mink-muskrat and paramecium-didinium data sets).

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“…In Section 2, we extend the BCF model to simulate external perturbations numerically. There are various prey-predator type competitive models with perturbations [16,17,18,19,20,21,22], however, most of them are with small, stochastic perturbations. The behaviors of conservation laws with external perturbations have been seldom considered.…”

Section: Introductionmentioning

confidence: 99%

Stability and restoration phenomena in competitive systems

Uechi

Akutsu

2013

Progress of Theoretical and Experimental Physics

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A conservation law and stability, recovering phenomena and characteristic patterns of a nonlinear dynamical system have been studied and applied to biological and ecological systems. In our previous study, we proposed a system of symmetric 2n-dimensional conserved nonlinear differential equations with external perturbations. In this paper, competitive systems described by 2-dimensional nonlinear dynamical (ND) model with external perturbations are applied to population cycles and recovering phenomena of systems from microbes to mammals.The famous 10-year cycle of population density of Canadian lynx and snowshoe hare is numerically analyzed. We find that a nonlinear dynamical system with a conservation law is stable and generates a characteristic rhythm (cycle) of population density, which we call the standard rhythm of a nonlinear dynamical system. The stability and restoration phenomena are strongly related to a conservation law and balance of a system. The standard rhythm of population density is a manifestation of the survival of the fittest to the balance of a nonlinear dynamical system. 1

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Prediction of predator–prey populations modelled by perturbed ODEs

Cited by 12 publications

References 17 publications

Simple testing procedures for the Holling type II model

Simple testing procedures for the Holling type II model

Likelihood ratio test for a special predator-prey system

Stability and restoration phenomena in competitive systems

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