Single-species Population Model Research Articles

Invariant manifolds for stochastic delayed partial differential equations of parabolic type

The aim of this paper is to prove the existence and smoothness of stable and unstable invariant manifolds for a stochastic delayed partial differential equation of parabolic type. The stochastic delayed partial differential equation is firstly transformed into a random delayed partial differential equation by a conjugation, which is then recast into a Hilbert space. For the auxiliary equation, the variation of constants formula holds and we show the existence of Lipschitz continuous stable and unstable manifolds by the Lyapunov–Perron method. Subsequently, we prove the smoothness of these invariant manifolds under appropriate spectral gap condition by carefully investigating the smoothness of auxiliary equation, after which, we obtain the invariant manifolds of the original equation by projection and inverse transformation. Eventually, we illustrate the obtained theoretical results by their application to a stochastic single-species population model.

Long transients in discontinuous time-discrete models of population dynamics

Traditionally, mathematical modelling of population dynamics was focused on long-term, asymptotic behaviour (systems attractors), whereas the effects of transient regimes were largely disregarded. However, recently there has been a growing appreciation of the role of transients both in empirical ecology and theoretical studies. Among the main challenges are identification of the mechanisms triggering transients in various dynamical systems and understanding of the corresponding scaling law of the transient’s lifetime; the latter is of a vital practical importance for long-term ecological forecasting and regime shifts anticipation. In this study, we reveal and investigate various patterns of long transients occurring in two generic time-discrete population models which are mathematically described by discontinuous (piece-wise) maps. In particular, we consider a single-species population model and a predator–prey system, in each model we assume that the dispersal of species at the end of each season is density dependent. For both models, we demonstrate transients due to crawl-by dynamics, chaotic repellers, chaotic saddles, ghost attractors, and a rich variety of intermittent regimes. For each type of transient, we investigate the corresponding scaling law of the transient’s lifetime. We explore the space of key model parameters, to find where particular types of long transients can be expected, and we show that long transients are omnipresent since they can be observed within a wide range of model parameters. We also reveal the possibility of complex patterns occurring as a cascade of transients of different types. We also considered a stochastic version of the model where some parameters exhibit random fluctuations. We show that stochasticity can reduce, extend or produce new patterns of long transients. We conclude that the discontinuity in population models significantly facilitates the emergence of long transients by creating new types and increasing parameter domains of the corresponding transient dynamics. Another important conclusion is that the asymptotic regime of population dynamics is hardly possible to predict based on a finite time course of species densities, which is crucial for ecosystem management and decision making.

Survival analysis of a stochastic impulsive single-species population model with migration driven by environmental toxicant

Considering the influence of environmental toxicant on population migration between patches, we propose and study a stochastic impulsive single-species population model with migration driven by environmental toxicant in this paper. We first discuss the existence and uniqueness of global positive solutions of the model by constructing the Lyapunov function. Then, we obtain sufficient conditions for extinction, stochastic persistence and persistence in the mean of the single-species population. Finally, we present some numerical simulations to illustrate our results. These results provide insights for the conservation and management of species in polluted environments.

Threshold dynamics of a stochastic single population model with Allee effect

A stochastic single species population model with Allee effect is considered in this paper. By using the theory of Random Dynamical System (RDS), the complete classification of the global dynamics of stochastic system is given. The theoretical results show that the dynamics of the system is completely determined by the threshold λ=2r−σ2: if λ≤0, the population will extinct, i.e., the solutions of the model will go to zero; if λ>0, the population is permanent. Moreover, the transition probabilities of the solutions weakly converge to the unique stationary distribution and the explicit density function of the stationary distribution is given.

Numerical oscillation of nonlinear generalized delay single species population model

Numerical oscillation of nonlinear generalized delay single species population model

Permanence for an impulsively perturbed periodic single-species model considering Monod-Haldane Type functional response as predation term

Evolution of species encounter different sudden short term changes in population density in this ever changing environment. Impulsive differential equations are an appropriate tool to model population ecology. A single-species population model under the influence of constant and linear impulsive perturbations at fixed moments of time is analyzed in which Monod-Haldane functional response is taken as predation term. The conditions required for the permanence of species are established by using comparison analysis technique. To substantiate the theoretical findings, numerical simulation is done using MATLAB.

Persistence Analysis of a Stochastic Single Species Population Model with Allee Effect

This paper considers a stochastic single species population model with the Allee effect. We mainly analyze the persistence of population, and obtain the strictly critical conditions of persistence and extinction of population: When [Formula: see text], the population will go extinct in probability; when [Formula: see text], the stochastic model possesses a unique stationary distribution which implies that the population is permanent. The obtained results are finally verified through numerical simulations.

A Slow Single-Species Model with Non-Symmetric Variation of the Coefficients

A single-species population model exhibiting a symmetric slow variation for the carrying capacity and intrinsic growth rate is evaluated explicitly. However, it is unrealistic to eliminate the possibility of a clear separation in the evolution of the biotic environmental elements; thus, this paper considers the situation where these elements have a hierarchical variation on the time scales. Accordingly, two particular situations are recognized, which are the carrying capacity varies faster than the growth rate and vice versa. Applying the multi-time scaling technique in such a system provides a small parameter, which leads us to construct analytical approximate expressions for the population behavior, using the perturbation approach. Such approximations display very good agreement with the numerical simulations.

Single-species population models with stage structure and partial tolerance in polluted environments.

We propose stage-structured single-species population models with psychological effects and partial tolerance in polluted environments in this paper. First, the conditions of extinction and the time for extinction are investigated respectively. Especially, the time for extinction takes longer as the value of the psychological effects increases. Then the weak persistence in the mean around the pollution-free equilibrium and the stochastic permanence have been derived under some moderate conditions. Further, the existence of a periodic solution for the periodic single-species population has been determined. The corresponding numerical simulations verify the efficiency of the main theoretical results.

Isolated periodic wave solutions arising from Hopf and Poincaré bifurcations in a class of single species model

In this paper, we consider the bifurcations of local and global isolated periodic traveling waves in a single species population model described by a reaction-diffusion equation. Based on the singular point quantity algorithm of conjugate symmetric complex systems, we investigate Hopf bifurcation from all equilibrium points for the corresponding planar traveling wave system. We obtain all center conditions and construct one perturbed Hamiltonian system to study Poincaré bifurcation. Further, using the Chebyshev criterion, we develop a utilized approach to prove the existence of at most two limit cycles in a piecewise continuous parameter interval. Finally, the existence of double isolated periodic traveling waves for the model is established, and the results are illustrated by numerical simulation. It is shown that in a population model with density-dependent migrations and Allee effect, two large amplitude oscillations (isolated periodic traveling waves) can exist simultaneously.

Qualitative Analysis of a Single-Species Model with Distributed Delay and Nonlinear Harvest

In this paper, a single-species population model with distributed delay and Michaelis-Menten type harvesting is established. Through an appropriate transformation, the mathematical model is converted into a two-dimensional system. Applying qualitative theory of ordinary differential equations, we obtain sufficient conditions for the stability of the equilibria of this system under three cases. The equilibrium A1 of system is globally asymptotically stable when br−c>0 and η<0. Using Poincare-Bendixson theorem, we determine the existence and stability of limit cycle when br−c>0 and η>0. By computing Lyapunov number, we obtain that a supercritical Hopf bifurcation occurs when η passes through 0. High order singularity of the system, such as saddle node, degenerate critical point, unstable node, saddle point, etc, is studied by the theory of ordinary differential equations. Numerical simulations are provided to verify our main results in this paper.

Hopf Bifurcation in a Delayed Single Species Network System

In this paper, we focus on a network system which describes spatiotemporal dynamics of single species population at different patches since species can have different features in various life stages and different behaviors in various spatial environments. With the effect of time delay and spatial dispersion, homogenous, periodic and spatiotemporally nonhomogeneous distributions are identified. The stability analysis is carried out for the discrete-space and continuous-time network on single species with time delay and the Hopf bifurcation of the single species population model in a network is explored. Formulas for determining the direction of Hopf bifurcation are derived by using the center manifold method and the normal form theorem. It is found that the network can generate spatial patterns only when time delay is present. Finally, numerical simulations are performed which agree well with our theoretical result, i.e. this discrete-space and continuous-time model admits regular temporal patterns since the delay induces Hopf bifurcations with network structure.

Evolutionary dynamics of single species model with Allee effects and aposematism

A single species population model subject to additive Allee effects, due to predation satiation is proposed and analyzed by considering a continuous strategy evolutionary game theory (EGT). Moreover, we consider the aposematic behavior of the population. We assume a single trait that affects aposematic behavior and saturation constant, following the Gaussian distribution. First, we analyze our single species deterministic model in global perspective. Then, we consider the aposematism parameter and saturation constant are functions of a mean phenotypic trait subject to evolution and formulate a model that describes the coupled population and mean trait dynamics. We can see that the extinction equilibrium also can be an evolutionary stable strategy (ESS) depending on the evolutionary functional forms. The ratio of variation in aposematic behavior and saturation constant phenotypes has some important role in the equilibria stability and ESS conditions of our evolutionary model. We find the ESS conditions for all equilibria and numerical simulation suggests that our evolutionary model can have multiple ESSs.

Evolutionary Dynamics of a Single-Species Population Model with Multiple Delays in a Polluted Environment

Evolutionary Dynamics of a Single-Species Population Model with Multiple Delays in a Polluted Environment

STUDY OF SINGLE-SPECIES POPULATION MODELS

In this research paper one dimensional population models developed centuries ago shows that growth and/decay of single homogeneous populations But environmental effects spatial heterogeneity or age-structure deterministic models prevailing single species population models.

Survival analysis of a single-species population model with fluctuations and migrations between patches

We formulate a single-species population model with fluctuations and migrations between two different patches (the nature reserve and the natural environment) to describe the situation that endangered species often meets with. Firstly, we prove that there exists a unique global positive solution to a stochastic single-species model with any positive initial value. Then, sufficient conditions that guarantee extinction and persistence in the mean of solutions are derived as the main results. It turns out that, under moderate fluctuation conditions, migration rates play significant roles for the persistence of endangered species in nature reserve and the natural environment. We further provide the sustainable strategies for local managers to avoid the extinction of endangered species, and separately carry out illustrative examples and numerical simulations at the end of this contribution.

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.

Superadditive and subadditive dynamics are not inherent to the types of interacting threat

Species and ecosystems usually face more than one threat. The damage caused by these multiple threats can accumulate nonlinearly: either subadditively, when the joint damage of combined threats is less than the damages of both threats individually added together, or superadditively, when the joint damage is greater than the two individual damages added together. These additivity dynamics are commonly attributed to the nature of the threatening processes, but conflicting empirical observations challenge this assumption. Here, we use a theoretical model to demonstrate that the additivity of threats can change with different magnitudes of threat impacts (effect of a threat on the population parameter, like growth rate). We use a harvested single-species population model to integrate the effects of multiple threats on equilibrium abundance. Our results reveal that threats do not always display consistent additive behavior, even in simple systems. Instead, their additivity depends on the magnitudes of the impacts of two threats, and the population parameter that is impacted by each threat. In our model specifically, when multiple threats have a low impact on the growth rate of a population, they display superadditive dynamics. In contrast, threats that impact the species’ carrying capacity are always additive or subadditive. These dynamics can be understood by reference to the curvature of the relationship between a given population parameter (e.g., growth) and equilibrium population size. Our results suggest that management actions can achieve amplified benefits if they target low-amplitude threats that affect the growth rate, since these will be in a superadditive phase. More generally, our results suggest that cumulative impact theory should focus more than previously on the magnitude of the impact on the population parameter, and should be cautious about attributing additive dynamics to particular threat combinations.

Local adaptation, dispersal evolution, and the spatial eco‐evolutionary dynamics of invasion

Local adaptation and dispersal evolution are key evolutionary processes shaping the invasion dynamics of populations colonizing new environments. Yet their interaction is largely unresolved. Using a single-species population model along a one-dimensional environmental gradient, we show how local competition and dispersal jointly shape the eco-evolutionary dynamics and speed of invasion. From a focal introduction site, the generic pattern predicted by our model features a temporal transition from wave-like to pulsed invasion. Each regime is driven primarily by local adaptation, while the transition is caused by eco-evolutionary feedbacks mediated by dispersal. The interaction range and cost of dispersal arise as key factors of the duration and speed of each phase. Our results demonstrate that spatial eco-evolutionary feedbacks along environmental gradients can drive strong temporal variation in the rate and structure of population spread, and must be considered to better understand and forecast invasion rates and range dynamics.

Long time behaviors of single-species population models with psychological effect and impulsive toxicant in polluted environments

In this paper, we formulate and study the single-species population models with impulsive toxicant input in polluted environments. For the deterministic single-species population model, the threshold between persistence and extinction is established for single-species population. For the stochastic single-species population model, there is a unique positive solution, which is a prerequisite for analyzing the long-term behavior of the stochastic model. Then, a threshold dynamic determined by μ̄ is established, which mainly determines the persistence or extinction of the single-species. Numerical simulations are carried out to support our theoretical results.