Optimal harvesting of diffusive models in a nonhomogeneous environment

Braverman, Elena; Braverman, L. M.

doi:10.1016/j.na.2009.04.025

Cited by 41 publications

(44 citation statements)

References 22 publications

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“…In addition, we extend the results of [9] to the case when the diffusion is not necessarily regular but the diffusion strategy is the same for both species, and it is different from [3] where, as the diffusion coefficient tends to infinity, the solution tends to the carrying capacity which coincides with the ideal free distribution. If this is the case, higher diffusion coefficient leads to a disadvantage in a competition.…”

Section: Introductionmentioning

confidence: 80%

“…If a 1 is space-independent, in the first equation of (2.2) we obtain the type of dispersal ∆(u/P ). In the particular case when P ≡ K (or P is proportional to K) we have the term ∆(u/K), first introduced in [3] and later considered in [10]- [13]. If Q is constant, while a 2 is proportional to 1/K, we have the dispersal type ∇ · ( 1 K ∇v) which was considered in [11,12].…”

Section: Dispersal Modelingmentioning

confidence: 99%

“…An ideal free distribution in a temporally constant but spatially heterogeneous environment is expected to be a solution of the system. There were several possible approaches to model dispersal in such a way that the ideal free distribution is a stationary solution of the equation, see, for example, [3,6,7,8].…”

Section: Introductionmentioning

confidence: 99%

“…The first goal is to unify different approaches to diffusion strategies and dispersal, for example, the advection to better environments in [5] and the model where not the population density but its ratio to locally available resources diffuses [3]. A type of dispersal which includes [3,5] is introduced in Section 2.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Competitive–cooperative models with various diffusion strategies

Braverman¹,

Kamrujjaman²

2016

Computers & Mathematics with Applications

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The paper is concerned with different types of dispersal chosen by competing species. We introduce a model with the diffusion-type term ∇·[a∇ (u/P )] which includes some previously studied systems as special cases, where a positive space-dependent function P can be interpreted as a chosen dispersal strategy. The well-known result that if the first species chooses P proportional to the carrying capacity while the second does not then the first species will bring the second one to extinction, is also valid for this type of dispersal. However, we focus on the case when the ideal free distribution is attained as a combination of the two strategies adopted by the two species. Then there is a globally stable coexistence equilibrium, its uniqueness is justified. If both species choose the same dispersal strategy, non-proportional to the carrying capacity, then the influence of higher diffusion rates is negative, while of higher intrinsic growth rates is positive for survival in a competition.

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Section: Introductionmentioning

confidence: 80%

Section: Dispersal Modelingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Competitive–cooperative models with various diffusion strategies

Braverman¹,

Kamrujjaman²

2016

Computers & Mathematics with Applications

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“…Braverman and Mamdani [9] considered the optimal harvesting problems of single-species models with impulsive and continuous harvesting effects. Braverman and Braverman [8] analyzed the optimal harvesting problems of singlespecies models with diffusion.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of a stochastic service-resource mutualism model with Lévy noises and harvesting

Wang¹,

Du²,

Liu³

2017

J. Nonlinear Sci. Appl.

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In this paper, we propose a stochastic service-resource mutualism model with Lévy noises and harvesting. Under some assumptions, we study several dynamical properties of the model. We first obtain the thresholds between persistence and extinction for both the service species and the resource species. Then we give sharp sufficient conditions for stability in distribution of the model. Finally, we establish sufficient and necessary criteria for the existence of the optimal harvesting policy. The optimal harvesting effort and maximum of sustainable yield are also obtained. Our results reveal that the persistence, extinction, stability in distribution and optimal harvesting strategy have close relationships with the random noises.

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On evolutionary stability of carrying capacity driven dispersal in competition with regularly diffusing populations

Korobenko

Braverman

2013

J. Math. Biol.

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Two competing populations in spatially heterogeneous but temporarily constant environment are investigated: one is subject to regular movements to lower density areas (random diffusion) while the dispersal of the other is in the direction of the highest per capita available resources (carrying capacity driven diffusion). The growth of both species is subject to the same general growth law which involves Gilpin-Ayala, Gompertz and some other equations as particular cases. The growth rate, carrying capacity and dispersal rate are the same for both population types, the only difference is the dispersal strategy. The main result of the paper is that the two species cannot coexist (unless the environment is spatially homogeneous), and the carrying capacity driven diffusion strategy is evolutionarily stable in the sense that the species adopting this strategy cannot be invaded by randomly diffusing population. Moreover, once the invasive species inhabits some open nonempty domain, it would spread over any available area bringing the native species diffusing randomly to extinction. One of the important technical results used in the proofs can be interpreted in the form that the limit solution of the equation with a regular diffusion leads to lower total population fitness than the ideal free distribution.

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Optimal harvesting of diffusive models in a nonhomogeneous environment

Cited by 41 publications

References 22 publications

Competitive–cooperative models with various diffusion strategies

Competitive–cooperative models with various diffusion strategies

Dynamics of a stochastic service-resource mutualism model with Lévy noises and harvesting

On evolutionary stability of carrying capacity driven dispersal in competition with regularly diffusing populations

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