Dynamics of Kolmogorov systems of competitive type under the telegraph noise

Du, Nguyen Huu; Dang, Nguyen Hai

doi:10.1016/j.jde.2010.08.023

Cited by 42 publications

(29 citation statements)

References 17 publications

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“…For each state e ∈ M, we denote by π e t (z 0 ) the solution of system (1.2) in the state e with the initial value z 0 ∈ K. As in [11], the Ω-limit set of the trajectory starting from an initial value z 0 ∈ K is defined by Ω(z 0 , ω) = T >0 t>T (S(t, ω, z 0 ), I(t, ω, z 0 ), R(t, ω, z 0 )). We here use the notation "Ω-limit set" in lieu of the usual one "ω-limit set" in the deterministic dynamical system for avoiding notational conflict with the element notation ω in the probability sample space.…”

Section: ω-Limit Set and Attractormentioning

confidence: 99%

“…Step 3. We now shall that the stationary distribution ν * of the process ((S(t), I(t), R(t)), r(t)) has a density f * with respect to the product measure m on X and supp(f * ) = Γ × M. Since the argument is similar to that of [11], we here only sketch the proof to point out the difference with it.…”

Section: )mentioning

confidence: 99%

“…It is worth mentioning that the modified method can be applied to the case where all parameters of the model considered in [7,15,16] were derived by the stochastic process r(t), however, here we only allow the transmission rate β of model (1.2) to be disturbed because in reality it may be more sensitive to environmental fluctuations than other parameters of model (1.1) for human populations. In particular, by the new method, under weaker conditions we can directly extend the results in [11,12,15] from two environmental states to any finite ones. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 98%

“…In the case of the disease persistence, we will analyze the ergodic behavior of system (1.2). By skilled techniques, authors in [11,12] applied the stochastic stability theory of Markov processes in [22,23,24,25] to two-dimensional Kolmogorov systems of competitive type switching between two environmental states. They obtained the sufficient conditions for the global attractivity of the Ω-limit set of the system and the convergence of the instantaneous measure to the stationary measure in total variation.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, the modified techniques avoid the limitations for both the number of environmental regimes and the dimension of the considered system. For instance, the method used by [11,12] does not deal with the case where the dimension of the system is higher than the number of environmental states of Markov chain, while the modified method can do.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching

Dan

Liu

Cui

2017

Journal of Differential Equations

116

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This paper studies the spread dynamics of a stochastic SIRS epidemic model with nonlinear incidence and varying population size, which is formulated as a piecewise deterministic Markov process. A threshold dynamic determined by the basic reproduction number R 0 is established: the disease can be eradicated almost surely if R 0 < 1, while the disease persists almost surely if R 0 > 1. The existing method for analyzing ergodic behavior of population systems has been generalized. The modified method weakens the required conditions and has no limitations for both the number of environmental regimes and the dimension of the considered system. When R 0 > 1, the existence of a stationary probability measure is obtained. Furthermore, with the modified method, the global attractivity of the Ω-limit set of the system and the convergence in total variation to the stationary measure are both demonstrated under a mild extra condition.

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Section: ω-Limit Set and Attractormentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching

Dan

Liu

Cui

2017

Journal of Differential Equations

116

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Stochastic Lotka–Volterra food chains

Hening

Nguyen

2017

J. Math. Biol.

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Abstract. We study the persistence and extinction of species in a simple food chain that is modelled by a Lotka-Volterra system with environmental stochasticity. There exist sharp results for deterministic Lotka-Volterra systems in the literature but few for their stochastic counterparts. The food chain we analyze consists of one prey and n − 1 predators. The jth predator eats the j − 1th species and is eaten by the j + 1th predator; this way each species only interacts with at most two other species -the ones that are immediately above or below it in the trophic chain. We show that one can classify, based on an explicit quantity depending on the interaction coefficients of the system, which species go extinct and which converge to their unique invariant probability measure. Our work can be seen as a natural extension of the deterministic results of Gard and Hallam '79 to a stochastic setting.As one consequence we show that environmental stochasticity makes species more likely to go extinct. However, if the environmental fluctuations are small, persistence in the deterministic setting is preserved in the stochastic system. Our analysis also shows that the addition of a new apex predator makes, as expected, the different species more prone to extinction.Another novelty of our analysis is the fact that we can describe the behavior the system when the noise is degenerate. This is relevant because of the possibility of strong correlations between the effects of the environment on the different species.

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The competitive exclusion principle in stochastic environments

Hening

Nguyen

2020

J. Math. Biol.

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In its simplest form, the competitive exclusion principle states that a number of species competing for a smaller number of resources cannot coexist. However, it has been observed empirically that in some settings it is possible to have coexistence. One example is Hutchinson's 'paradox of the plankton'. This is an instance where a large number of phytoplankton species coexist while competing for a very limited number of resources. Both experimental and theoretical studies have shown that temporal fluctuations of the environment can facilitate coexistence for competing species. Hutchinson conjectured that one can get coexistence because nonequilibrium conditions would make it possible for different species to be favored by the environment at different times.In this paper we show in various settings how a variable (stochastic) environment enables a set of competing species limited by a smaller number of resources or other density dependent factors to coexist. If the environmental fluctuations are modeled by white noise, and the percapita growth rates of the competitors depend linearly on the resources, we prove that there is competitive exclusion. However, if either the dependence between the growth rates and the resources is not linear or the white noise term is nonlinear we show that coexistence on fewer resources than species is possible. Even more surprisingly, if the temporal environmental variation comes from switching the environment at random times between a finite number of possible states, it is possible for all species to coexist even if the growth rates depend linearly on the resources. We show in an example (a variant of which first appeared in Benaim and Lobry '16) that, contrary to Hutchinson's explanation, one can switch between two environments in which the same species is favored and still get coexistence.2010 Mathematics Subject Classification. 92D25, 37H15, 60H10, 60J05, 60J99.

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Dynamics of Kolmogorov systems of competitive type under the telegraph noise

Cited by 42 publications

References 17 publications

Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching

Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching

Stochastic Lotka–Volterra food chains

The competitive exclusion principle in stochastic environments

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