Conditions For Extinction Research Articles

Persistence and extinction for stochastic delay differential model of prey predator system with hunting cooperation in predators

Stochastic differential models provide an additional degree of realism compared to their corresponding deterministic counterparts because of the randomness and stochasticity of real life. In this work, we study the dynamics of a stochastic delay differential model for prey–predator system with hunting cooperation in predators. Existence and uniqueness of global positive solution and stochastically ultimate boundedness are investigated. Some sufficient conditions for persistence and extinction, using Lyapunov functional, are obtained. Illustrative examples and numerical simulations, using Milstein’s scheme, are carried out to validate our analytical findings. It is observed that a small scale of white noise can promote the survival of both species; while large noises can lead to extinction of the predator population.

Ergodic stationary distribution and extinction of a stochastic SIRS epidemic model with logistic growth and nonlinear incidence

A stochastic SIRS model with logistic growth and nonlinear incidence rate is probed in this paper. We exemplify that the proposed stochastic SIRS model reveals a global and positive solution. By applying suitable Lyapunov functions, the sufficient conditions for the existence of ergodic stationary distribution of the solution to the stochastic SIRS model are derived. Furthermore, we acquire the sufficient conditions for extinction of the infectious disease.

Dynamics of a multigroup SIQS epidemic model under regime switching

In this paper, we study the dynamical behavior of a multigroup SIQS epidemic model, which is formulated as a piecewise deterministic Markov process. Sufficient conditions for extinction and persistence in the mean of the diseases are obtained. In addition, in the case of diseases persistence, we establish sufficient conditions for the existence of positive recurrence of the solutions to the model by using the stochastic Lyapunov function with regime switching.

Stationary distribution of stochastic NP ecological model under regime switching

Marine ecosystem is inevitably disturbed by random factors. This article discusses a nutrient–phytoplankton (NP) model with TPP which is disturbed by both white and telegraph noises. Firstly, we get sufficient conditions for the existence of ergodic stationary distribution by constructing a suitable stochastic Lyapunov function with regime switching. And then we establish sufficient conditions for extinction of the phytoplankton. Finally, the results of this paper are verified by numerical simulation.

Dynamical behavior of a stochastic predator-prey model with stage structure for prey

In the present paper, we focus on a stochastic predator-prey model with stage structure for prey. Firstly, by using the stochastic Lyapunov function method, we obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Then we establish sufficient conditions for extinction of the predator population in two cases. Some examples and numerical simulations are carried out to validate our analytical findings.

From nest site lottery to host lottery: continuous model of growth suppression driven by the availability of nest sites for newborns or hosts for parasites and its impact on the selection of life history strategies.

The idea that selection works in different ways during free population growth and at the equilibrium population size has been present in theoretical biology for a long time. It was first expressed as an r and K selection concept and later clarified in the debate on fitness measures in life history theory. The latest discussion related to this topic is focused on the nest site lottery mechanism and the resulting new population growth model. In this mechanistic biphasic model, the suppression of growth is induced by a shortage of free nest sites for newborns. Before it occurs, the population can grow exponentially. In this paper, the continuous version of the model and its selective properties are analysed. We show a continuous smooth transition between different fitness measures operating during the exponential growth and suppressed growth phase and at the equilibrium population size. Then, the model is extended to the case of a population of parasites, where a constant number of nest sites is replaced by the dynamics of a population of their hosts, in the role of the limiting supply. Parasite strategies are selected under exponential and suppressed growth phases of the population of hosts. Transitions between different fitness measures and conditions for extinction of hosts by parasites are analysed. An interesting result is the possibility of a continuum of fitness measures of parasites for the unsuppressed exponential growth of the host population.

Existence and asymptotic behaviour of positive solutions to a stochastic multispecies Holling type II model

In this paper, we develop a stochastic Holling type II model from the commonly used deterministic Holling type II model by introducing Brownian motion as a stochastic perturbation, which can naturally account for the external environmental effects on a multispecies community. We study the existence and uniqueness of positive solutions to the stochastic model, and we establish sufficient conditions for strong persistence and extinction.

Analysis of a Stochastic Ratio-Dependent Predator-Prey System with Markovian Switching and Lévy Jumps

In this paper, the dynamics of a stochastic ratio-dependent predator-prey system with markovian switching and Lévy noise is studied. Firstly, we show the existence condition of global positive solution under the given positive initial value. Secondly, sufficient conditions for system extinction and persistence are obtained through some assumptions. Then, the sufficient conditions of stochastically persistence are obtained by combining stochastic analysis technique and M-matrix analysis. In addition, under appropriate conditions, we demonstrate the existence of a unique stationary distribution for a system without Lévy jumps. Finally, the empirical and Mlistein methods are used to verify the theoretical results through numerical simulation.

Assessing the viability of tri-trophic food chain model in designing a conservation plan: The case of dwindling Quokka population

The extinction and persistence dynamics of quokka (Setonix brachyurus) population at northern jarrah forest of Australia is investigated using mathematical modelling. Predator's management demands a comprehensive understanding of the ecological circumstances associated with predation. Predation by red foxes (Vulpes vulpes) has a significant role in reducing the native animal population. This paper mainly focuses on the extinction dynamics of quokka population and its reduction by red foxes, by qualitative and quantitative analyses of a designed tri-trophic food-chain model composing a prey (quokka), a mesopredator (red fox) and apex predator (dingo). Existence of solution are analysed and shown to be uniformly bounded. We applied the concept of basic reproduction number from epidemiology to the food chain model, to derive a condition for extinction and persistence of predator population. Global stability of the predator-free equilibrium is established by geometric approach. We use Partial Rank Correlation Coefficient (PRCC) for performing global sensitivity analysis to identify most influential model parameter responsible for quokkas prevalence and mortality. Results of numerical simulation for both deterministic and stochastic model confirms the analytical finding and support those of previous studies. The outcome of this work shows that persistence and existence of quokka depend on the demographic impacts of environmental stochasticity on its own population. Based on our results, many conservation strategies are suggested to improve the overall growth of the species. We believe that declination of red fox and the presence of dingoes are important for preserving the uniform occurrence of quokkas.

Survival analysis of single-species population diffusion models with chemotaxis in polluted environment

In this paper, single-species population diffusion models with chemotaxis in polluted environment are proposed and studied. For the deterministic single-species population diffusion model, the sufficient conditions for the extinction and strong persistence of the single-species population are established. For the stochastic single-species population diffusion model. First, we show that system has unique global positive solution. And then, the sufficient conditions for extinction and strongly persistent in the mean of the single-species are obtained. Numerical simulations are used to confirm the efficiency of the main results.

PERIODIC SOLUTION OF A STOCHASTIC SIQR EPIDEMIC MODEL INCORPORATING MEDIA COVERAGE

In this paper, we propose a stochastic SIQR epidemic model with periodic parameters and media coverage. Firstly, we study that the stochastic non-autonomous periodic system has a unique global positive solution. Secondly, by using the Khasminskii's theory, we prove that this stochastic periodic system has a nontrivial positive periodic solution. Then, we obtain the sufficient condition for extinction of the disease. Finally, numerical simulations are employed to illustrate our theoretical analysis.

Global dynamics analysis of a stochastic SIRS epidemic model with vertical transmission and different periods of immunity

In this work, we analyse a stochastic susceptible-infected-recovered-susceptible (SIRS) epidemic model with vertical transmission and different periods of immunity. This model has a global positive solution. Firstly, we establish sufficient conditions for extinction and persistence in the mean of a disease. Then, we prove the global stability of the system under a suitable condition of perturbation intensity. In the case of the non-autonomous system, we show that there exists at least one positive periodic solution. Finally, some numerical examples are introduced to show the validity of our results.

Extinction and stationary distribution of a competition system with distributed delays and higher order coupled noises.

A stochastic two-species competition system with saturation effect and distributed delays is formulated, in which two coupling noise sources are incorporated and every noise source has effect on two species' intrinsic growth rates in nonlinear form. By transforming the two-dimensional system with weak kernel into an equivalent four-dimensional system, sufficient conditions for extinction of two species and the existence of a stationary distribution of the positive solutions to the system are obtained. Our main results show that the two coupling noises play a significant role on the long time behavior of system.

Threshold behavior in two types of stochastic three strains influenza virus models

In this paper, we study two types of stochastic models consisting of three strains of influenza. For each type of the stochastic model, we establish sufficient conditions for extinction and persistence in the mean of these diseases by using the stochastic Lyapunov analysis method. Meanwhile, we obtain the threshold between persistence in the mean and extinction of the diseases.

Stationary distribution and extinction of a stochastic tuberculosis model

In this paper, we focus on a stochastic tuberculosis model. We first obtain the existence of a stationary distribution to the positive solutions by stochastic Lyapunov function method. Then we establish sufficient conditions for extinction of the disease. Finally, some examples and numerical simulations are provided to illustrate our theoretical results.

Persistence and extinction of a nonautonomous switching single-species population model

A novel nonautonomous switching single-species population model is proposed and investigated. The sufficient conditions for persistence and extinction of the solutions of the model are obtained. Moreover, corresponding results are given for periodic switching system.

Survivability and stochastic bifurcations for a stochastic Holling type II predator-prey model

In this paper, the dynamic properties of a stochastic model are studied through the stability of ergodic invariant measures on invariant sets. The threshold analysis of strong stochastic persistence and extinction is given. Moreover, the necessary and sufficient condition for persistence and extinction in the sense of time average is give for a special critical state. The stochastic bifurcation phenomenon of the model is studied from the viewpoint of dynamic bifurcation. The main conclusions are verified by examples and numerical simulations. In addition, the intra-specific competition of two species are considered in this paper, the importance of intra-specific competition is also illustrated by theoretical results. The results can also provide a theoretical basis for the modeling of stochastic population models.

On a degenerate nonlocal parabolic equation with variable source

We study the Dirichlet problem for the degenerate nonlocal parabolic equationut−div(a(‖∇u‖L2(Ω)2)∇u)=b(‖u‖L2(Ω)2)|u|q(x,t)−2u+fin QT, where QT:=Ω×(0,T), T>0, Ω⊂Rn, n≥2, is a bounded domain with a sufficiently smooth boundary, a, b are real-valued functions defined on R+ and q(x,t) is a measurable function in QT with values in an interval [q−,q+]⊂(1,∞). We show that under appropriate conditions on a, b, q the problem has a global in time solution and find conditions of uniqueness. Sufficient conditions for extinction in finite time or strict positivity of ‖u(t)‖L2(Ω) for all t>0 are found, in the case f=0 the decay rates for ‖u(t)‖L2(Ω) are derived.

Threshold behavior in a stochastic SIR epidemic model with Logistic birth

In this paper, we consider a stochastic SIR epidemic model with Logistic birth. By using the stochastic Lyapunov function method, we show that the stochastic basic reproduction number R0S can be used to determine the threshold dynamics of the stochastic system. If R0S>1, we establish sufficient conditions for the existence of a stationary distribution of the positive solutions to the model. While if R0S<1, under some extra conditions, we obtain sufficient conditions for extinction of the disease. Finally, some examples and numerical simulations are provided to illustrate the theoretical results.

Fear factor in a prey–predator system in deterministic and stochastic environment

In this article, a prey–predator model has been studied where apart from direct predation the prey population is affected by the fear induced from predators. The basic reproduction of the prey population is reduced as a cost of fear. Biologically well posedness of the model system has been shown through positivity and boundedness of solutions. Existence criterion and stability analysis of the non-negative equilibrium points have been discussed and necessary conditions for uniform persistence have been derived for corresponding non-autonomous system. Also, sufficient conditions for the existence of positive periodic solution of the non-autonomous system have been established by utilizing the coincidence degree theorem. We further show that the positive periodic solution is global attractor under certain conditions. Next, we construct the stochastic model based upon the deterministic setup by perturbing the intrinsic growth rate of prey and natural mortality rate of predators. It has been shown that the stochastic system admits unique positive global solution initiates from anywhere in the interior of the positive quadrant. The sufficient conditions for extinction, non-persistence and weakly persistence of both the species have been derived along with the stochastic permanence of the system. We have verified our analytical results and make a comparison between the deterministic and stochastic setup by exhaustive numerical simulations.