Stability and explicit stationary density of a stochastic single-species model

Ji, Weiming; Hu, Guixin

doi:10.1016/j.amc.2020.125593

Cited by 5 publications

(7 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 6. This paper reports the global stability of a single-species ecosystem, while the stability in [3] did not involve the global one. This means that the stability in [3] depends heavily on the choice of initial value, while the global stability does not need such a choice.…”

Section: Resultsmentioning

confidence: 99%

“…Here, Z(t) is the population's quantity or density at the time t, and K and R > 0 are the intrinsic growth rate of the environmental capacity and the population. Because all the solutions of nonlinear ecosystems are difficult to provide accurately, people pay more attention to the long-term dynamic trend of an ecosystem, i.e., the long-term trend of population density (see, for example, [1][2][3][4][5]). People especially want to know whether the population will tend to a positive constant after a long time, which is related to the final long-term existence of the population.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Global Stabilization of a Single-Species Ecosystem with Markovian Jumping under Neumann Boundary Value via Laplacian Semigroup

Rao

Jia-lin²,

Yang

2021

Mathematics

View full text Add to dashboard Cite

By applying impulsive control, this work investigated the global stabilization of a single-species ecosystem with Markovian jumping, a time delay and a Neumann boundary condition. Variational methods, a fixed-point theorem, and Laplacian semigroup theory were employed to derive the unique existence of the global stable equilibrium point, which is a positive number. Numerical examples illuminate the feasibility of the proposed methods.

show abstract

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Global Stabilization of a Single-Species Ecosystem with Markovian Jumping under Neumann Boundary Value via Laplacian Semigroup

Rao

Jia-lin²,

Yang

2021

Mathematics

View full text Add to dashboard Cite

show abstract

“…Different from [2,3], the population density boundedness of the species was considered in this study due to the important factor (Remark 4 for details). In Example 1, authors assume N 0 � 2 and M 0 � 10, and we can see it from Figures 1-3 that 2 < u < 10.…”

Section: Remarkmentioning

confidence: 99%

“…where a > 0 and b > 0 describe the growth rate and the intraspecific competition; α > 0 measures the intensity of the environmental disturbances. In recent years, model (2) has been widely adopted in many applications (e.g., [3][4][5]). A large number of facts have shown that the spatial scale and structure of the environment can affect population interaction [6,7] and community composition [8].…”

Section: Introductionmentioning

confidence: 99%

Existence, Uniqueness, and Input‐to‐State Stability of Ground State Stationary Strong Solution of a Single‐Species Model via Mountain Pass Lemma

2021

View full text Add to dashboard Cite

In this study, the authors utilize mountain pass lemma, variational methods, regularization technique, and the Lyapunov function method to derive the unique existence of the positive classical stationary solution of a single-species ecosystem. Particularly, the geometric characteristic of saddle point in the mountain pass lemma guarantees that the equilibrium point is the ground state stationary solution of the ecosystem. Based on the obtained uniqueness result, the authors use the Lyapunov function method to derive the globally exponential stability criterion, which illuminates that under some suitable conditions, a certain internal competition is conducive to the global stability of the population, and a certain amount of family planning is conducive to the overall stability of the population. Most notably, the regularity technique of weak stationary solution employed in this study can also be applied to some existing literature related with time-delays reaction-diffusion systems for the purpose of regularization of weak solutions. Finally, an illuminative numerical example shows the effectiveness of the proposed methods.

show abstract

“…Because all solutions of nonlinear ecosystem are difficult to be given accurately, people pay more attention to the long-term dynamic trend of ecosystem, i.e., the long-term trend of population density (see, e.g. [1][2][3][4][5]). People especially want to know whether the population will tend to a positive constant after a long time, which is related to the final long-term existence of the population.…”

Section: Introductionmentioning

confidence: 99%