Estimating Predator-Prey Systems via Ordinary Differential Equations With Closed Orbits

Froda, Sorana; Colavita, Gino

doi:10.1111/j.1467-842x.2005.00388.x

Cited by 9 publications

(7 citation statements)

References 6 publications

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“…Future research should further exploit the periodic behavior on the limit cycle and incorporate all available data in inference (see assumptions in Froda and Colavita 2005), in a kind of two-step procedure. 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.…”

Section: Discussionmentioning

confidence: 99%

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%

Simple testing procedures for the Holling type II model

Froda

Zahedi

2009

Theor Ecol

Self Cite

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“…Further, note that the parameters (γ , β) do not appear explicitly in the means of the random variables defined in the model (3). The main originality of this paper consists in using the reparametrization proposed in Theorem A.1 (see Appendix A.1), to reduce the testing problem (2) to a familiar problem in testing hypotheses.…”

Section: Assumption (C 2 )mentioning

confidence: 98%

“…where ( x 0 , y 0 ) is chosen on the closed curve corresponding to the system (1), in accordance with the method given in Froda and Colavita [3]. By some computations, we get preliminary estimates of (u 0 , v 0 ) as given in Table 2.…”

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

confidence: 98%

“…So far, by using the similar model, Froda and Colavita [3] and Nkurunziza [4] as well as Froda and Nkurunziza [5] proposed estimation methods for the four parameters α, β, γ , δ. However, in some circumstances, it is of interest to know if the parameters are different from one geographical area to another or if they change over time.…”

Section: S Nkurunzizamentioning

confidence: 99%

“…In other words, the mink-muskrat is a predator-prey pair, which seems to satisfy the Downloaded by [University of California, San Francisco] at 08:41 04 December 2014 requirement that the muskrat is the main food supply for the predator. Also, the same data has been used in Froda and Colavita [3], in Nkurunziza [4] as well as in Froda and Nkurunziza [5] to illustrate their estimation methods.…”

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

confidence: 99%

See 2 more Smart Citations

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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Prediction of predator–prey populations modelled by perturbed ODEs

Froda

Nkurunziza

2006

J. Math. Biol.

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In this paper we explore a stochastic model in continuous time for predator-prey interactions, which accounts for the periodical behaviour observed in many animal populations. More precisely, we consider a solution to the classical Lotka-Volterra system of equations, but we view the actual population sizes as random perturbations of the solutions to this ODE system. Namely, we assume that the perturbations follow correlated Ornstein-Uhlenbeck processes; this approach generalizes the one of Froda and Colavita [Aust N Z J Stat 2:235-254, 2005] who considered only i.i.d. errors. This type of perturbed deterministic model allows to perform parameter estimation and to predict population sizes at future times. On the other hand, the present model refines the previous one since it takes into account the variability due to external factors and the time dependence in the random component. Moreover, this more flexible model improves the predictions of population sizes at future times. In order to illustrate this last point, we analyse two data sets.

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Estimating Predator-Prey Systems via Ordinary Differential Equations With Closed Orbits

Cited by 9 publications

References 6 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

Prediction of predator–prey populations modelled by perturbed ODEs

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