Effect of population size in a predator–prey model

Campillo, Fabien; Lobry, Claude

doi:10.1016/j.ecolmodel.2012.07.015

Cited by 23 publications

(28 citation statements)

References 15 publications

(23 reference statements)

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“…In this section, we describe known exact and approximate simulation methods for the simulation of the solution Z N of equation (1). To this end, we fix N ; with the notations ν j = hj N and a j (z) = N β j (z), equation (1) can be rewritten as…”

Section: Simulation Of Z Nmentioning

confidence: 99%

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Numerical methods in the context of compartmental models in epidemiology

2015

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Abstract. We consider compartmental models in epidemiology. For the study of the divergence of the stochastic model from its corresponding deterministic limit (i.e., the solution of an ODE) for long time horizon, a large deviations principle suggests a thorough numerical analysis of the two models. The aim of this paper is to present three such motivated numerical works. We first compute the solution of the ODE model by means of a non-standard finite difference scheme. Next we solve a constraint optimization problem via discrete-time dynamic programming: this enables us to compute the leading term in the large deviations principle of the time of extinction of a given disease. Finally, we apply the τ -leaping algorithm to the stochastic model in order to simulate its solution efficiently. We illustrate these numerical methods by applying them to two examples.Résumé. On considère des modèles comportementaux enépidémiologie. Afin d'étudier l'écart en temps long entre le modèle stochastique et sa limite loi des grands nombres (qui est la solution d'une EDO), on se base sur un principe des grandes dáviations, qui nous conduità mener uneétude numérique des deux modèles, sur trois aspects différents. Tout d'abord, nous calculons une solution approchée de l'EDOà l'aide d'une méthode numérique dite "non-standard". Ensuite une résolvons numériquement un problème de contrôle sous contrainte, afin de calculer le terme principal des grandes déviations du temps de sortie d'une situation endémique. Enfin nous mettons en oeuvre l'algorithme du "τ -leaping" pour simuler efficacement la solution du système stochastique. Nous illustrons ces simulations numériques en les appliquantà deux exemples.

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Section: Simulation Of Z Nmentioning

confidence: 99%

“…Recently, it has been shown (see [1]) that in some cases the ODE models can diverge from the corresponding stochastic models even for larger population sizes (N > 10 6 ).…”

Section: Introductionmentioning

confidence: 99%

Numerical methods in the context of compartmental models in epidemiology

2015

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“…Noise impacted models are widely used in almost all disciplines of natural science such as biology, physics, chemistry, etc. (Campillo, Joannides, & Larramendy-Valverde, 2011;Campillo & Lobry, 2012;Caraballo, 1990;Imhof & Walcher, 2005;Sakthivel & Luo, 2009) and are of interest here as we handle a logistic model driven by random noise. We wish to investigate whether there exist optimal fishing times analogous to those derived from the deterministic version model (1).…”

Section: The Modelmentioning

confidence: 99%

Optimal threshold density in a stochastic resource management model with pulse intervention

Tan

Ning

Tang

et al. 2019

Natural Resource Modeling

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Human activities and agricultural practices are having huge impacts on the development of fishery and land resources through different ways. To model such systems that involve harvesting, an impulsive model of natural resources with a stochastic noise perturbation element is formulated to study the relationship between (a) the maximal expectation of biomass after harvesting or fishing events and (b) the minimal expectation of pest biomass and the number of times pesticide is applied. Using a detailed analytical treatment, time estimation, and numerical demonstrations, we establish that the proposed mechanism is capable of maximizing fish populations at the end of a fishing season and minimizing pest numbers after a crop harvesting season once the intensity of the noise is relatively small. Investigations of the effects of different parameters reveal that theoretical predictions from the new stochastic model accord with those from the deterministic case. Recommendations for Resource Managers Various measures can be implemented to manage natural resources, such as adjusting fishing quantity and intensity to maximize fish population. In the natural environment, population growth is inevitably affected by the environment noise. So it is important to understand the noise effect to maintain sustainability of resources. Investigated methods are useful to converse resources and can be widely applied to control pests.

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“…But how many individuals is enough to be large depends on the model and its parameters. Even if deterministic models can give suitable results for only a hundred individuals, a unit choice of 10 3 (or even 10 6 ) may not be enough to avoid atto-fox type problem [11] (that is having 10 −18 individuals able to avoid extinction).…”

Section: Galton Watson Process: a Stochastic Matrix Population Modelmentioning

confidence: 99%

Galton-Watson Process and Bayesian Inference: A Turnkey Method for the Viability Study of Small Populations

Bertrand¹,

Daufresne²,

Kerioui³

et al. 2019

Preprint

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1 Sharp prediction of extinction times is needed in biodiversity monitoring and conservation management. 2 The Galton-Watson process is a classical stochastic model for describing population dynamics. Its evolution is like the matrix population model where offspring numbers are random. Extinction probability, extinction time, abundance are well known and given by explicit formulas. In contrast with the deterministic model, it can be applied to small populations.3 Parameters of this model can be estimated through the Bayesian inference framework. This enables to consider nonarbitrary scenarios. 4 We show how coupling Bayesian inference with the Galton-Watson model provides several features: i) a flexible modelling approach with easily understandable parameters ii) compatibility with the classical matrix population model (Leslie type model) iii) A non-computational approach which then leads to more information with less computing iv) a non-arbitrary choice for scenarios, parameters... It can be seen to go one step further than the classical matrix population model for the viability problem. 5 To illustrate these features, we provide analysis details for two examples whose one of which is a real life example.

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Effect of population size in a predator–prey model

Cited by 23 publications

References 15 publications

Numerical methods in the context of compartmental models in epidemiology

Numerical methods in the context of compartmental models in epidemiology

Optimal threshold density in a stochastic resource management model with pulse intervention

Galton-Watson Process and Bayesian Inference: A Turnkey Method for the Viability Study of Small Populations

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