Extinction State Research Articles

Propagation dynamics of a three-species predator–prey model with non-local dispersal in a shifting habitat

This paper is concerned with the propagation phenomenon of a non-local dispersal predator–prey system involving two preys and one predator in a shifting habitat. Assuming that the growth rate of each prey is moving to the right at a speed of [Formula: see text] and the habitat is suitable for each prey to survive before the shifting edge, while the habitat behind the boundary is not suitable for survival. For any given speed of the shifting habitat edge, we establish four types of forced extinction waves, which describe the conversion from the state of a saturated aboriginal prey with a pair of invading alien predator–prey, two aboriginal co-existent preys with an invading alien predator, a pair of aboriginal co-existent predator–prey and an invading alien prey, and the co-existence of three species to the extinction state, respectively. Meanwhile, we show the non-existence of some forced waves.

Mega El Niño instigated the end-Permian mass extinction.

The ultimate driver of the end-Permian mass extinction is a topic of much debate. Here, we used a multiproxy and paleoclimate modeling approach to establish a unifying theory elucidating the heightened susceptibility of the Pangean world to the prolonged and intensified El Niño events leading to an extinction state. As atmospheric partial pressure of carbon dioxide doubled from about 410 to about 860 ppm (parts per million) in the latest Permian, the meridional overturning circulation collapsed, the Hadley cell contracted, and El Niños intensified. The resultant deforestation, reef demise, and plankton crisis marked the start of a cascading environmental disaster. Reduced carbon sequestration initiated positive feedback, producing a warmer hothouse and, consequently, stronger El Niños. The compounding effects of elevated climate variability and mean state warming led to catastrophic but diachronous terrestrial and marine losses.

Stationary distribution and mean extinction time in a generalist prey–predator model driven by Lévy noises

In this study, a delayed generalist prey–predator model incorporating the logistic-like growth and Holling type-III functional response for predator subjected to two Lévy noises is established. Such environmental fluctuations can describe abrupt change in population density. First, we conduct stability analysis and determine the critical value of time delay when Hopf bifurcation takes place. Then, the existence of the global positive solution and stability in distribution of stochastic model are studied. Numerical results are given to illustrate the analytical findings, and mainly dedicated to the consequences of time delay and Lévy noises on the stationary distribution, mean extinction time and reproduction rate of prey–predator populations, via stationary probability distribution function (PDF), mean first passage time (MFPT) and relaxation time, respectively. Our findings indicate that the decreases in time delay, noise intensities (Di,i=1,2), and stability indexes within the range of αi∈[1,2] are more likely to maintain the population in a high density state. Phase transition occurs when the system is in the suitable parameter regime. Additionally, there exist optimal noise intensities that slow down switching of prey species from a stable state to an extinction state. Smaller time delay τ and stability indexes αi inhibit population extinction, whereas smaller skewness parameter β speed up prey extinction. More importantly, two noises can tune the enhanced stability characteristic of the system.

The impact of water storage capacity on plant dynamics in arid environments: A stoichiometric modeling approach

Plants in arid environments have evolved many strategies to resist drought. Among them, the developed water storage tissue is an essential characteristic of xerophytes. To clarify the role of water storage capacity in plant performance, we originally formulate a stoichiometric model to describe the interaction between plants and water with explicit water storage. Via an ecological reproductive index, we explore the effects of precipitation and water storage capacity on plant dynamics. The model possesses saddle–node bifurcation and forward or backward bifurcation, and the latter may lead to the emergence of alternative stable states between a stable survival state and a stable extinction state. Numerical simulations illustrate the persistence and resilience of plants regulated by soil conditions, precipitation and water storage capacity. Our findings contribute to the botanical theory in the perspectives of environmental change and plant water storage traits.

Assessing mutualistic metacommunity capacity by integrating spatial and interaction networks

We develop a spatially realistic model of mutualistic metacommunities that exploits the joint structure of spatial and interaction networks. Assuming that all species have the same colonisation and extinction parameters, this model exhibits a sharp transition between stable non-null equilibrium states and a global extinction state. This behaviour allows defining a threshold on colonisation/extinction parameters for the long-term metacommunity persistence. This threshold, the ‘metacommunity capacity’, extends the metapopulation capacity concept and can be calculated from the spatial and interaction networks without needing to simulate the whole dynamics. In several applications we illustrate how the joint structure of the spatial and the interaction networks affects metacommunity capacity. It results that a weakly modular spatial network and a power-law degree distribution of the interaction network provide the most favourable configuration for the long-term persistence of a mutualistic metacommunity. Our model that encodes several explicit ecological assumptions should pave the way for a larger exploration of spatially realistic metacommunity models involving multiple interaction types.

Traveling wave fronts in a single species model with cannibalism and strongly nonlocal effect

<p>In this paper we studied traveling front solutions of a single species model with cannibalism and nonlocal effect. For a particular class of kernels, the existence of traveling front solutions connecting the extinction state with the positive equilibrium was established for the strongly nonlocal effect case. Our approach was to reformulate it as a singular perturbed problem, and then tackle this problem by using dynamical systems techniques, in particular, geometric singular perturbation theory and Fenichel's invariant manifold theory.</p>

Propagation dynamics of nonlocal dispersal competition systems in time-periodic shifting habitats

This paper is concerned with the propagation dynamics of the time periodic Lotka-Volterra competition systems with nonlocal dispersal in a shifting habitat. We first obtain three types of time-periodic forced waves connecting the extinction state to the co-existence state, itself and the semi-trivial state, which describe the conversion from the state of two aboriginal co-existent competing species, two invading alien competitors, and a saturated aboriginal competitor with another invading alien competitor to the extinction state, respectively. This provides a comprehensive explanation of the point-wise extinction dynamics of these two competing species under such a time-periodic worsening habitat. Then, we establish the spreading properties of the associated Cauchy problem depending on the range of the shifting speed. More specifically, we give a complete description on the threshold values for the extinction as well as persistence (by moving with asymptotic speed). Our results reveal the possibility that a competitively weaker species with a much faster spreading speed can drive a competitively stronger species with a slower spreading speed to extinction. The discussion in this paper applies to both cases of weak competition and strong-weak competition. In particular, we need to point out that some combined effects of nonlocal dispersal, two-variable coupling and time-periodic shifting heterogeneity in this system pose extra difficulties in mathematical treatment, which are dealt with by introducing new approaches.

Spatial propagation of a lattice predation–competition system with one predator and two preys in shifting habitats

AbstractThis paper deals with a discrete diffusive predator–prey system involving two competing preys and one predator in a shifting habitat induced by the climate change. By applying Schauder's fixed‐point theorem on various invariant cones via constructing several pairs of generalized super‐ and subsolutions, we establish four different types of supercritical and critical forced extinction waves, which describe the conversion from the state of a saturated aboriginal prey with a pair of invading alien predator–prey, two competing aboriginal coexistent preys with an invading alien predator, a pair of aboriginal coexistent predator–prey and an invading alien prey, and the coexistence of three species to the extinction state, respectively. Meanwhile, the nonexistence of some subcritical forced waves is showed by contradiction. Furthermore, some numerical simulations are given to present and promote the theoretical results.

Continuity of the spectral radius, applied to structured semelparous two-sex population models

If a structured discrete-time population model x n = F ( x n − 1 ) , n ∈ N , takes account of two-sexes, the year-to-year turnover operator F often has a first-order approximation B that is homogeneous and order-preserving rather than linear and positive. Still, one can define the spectral radius T = r ( B ) , the basic turnover number, which plays the role of a threshold parameter between extinction and persistence. If T < 1 , the extinction state is locally stable, and if T > 1 the population shows various degrees of persistence that depend on the irreducibility properties of the dispersal kernels. This threshold property raises interest in the continuous dependence of the basic turnover number on the parameters in the population model. In this paper, we concentrate on the continuous dependence of T = r ( B ) on mating functions.

Monitoring and Characterizing the Flame State of a Bluff-Body Stabilized Burner by Electrical Capacitance Tomography

Unstable combustion phenomena such as flame flashback, flame liftoff, extinction and blowout frequently take place during the operation of the bluff-body stabilized burner. Therefore, flame state monitoring is necessary for the safe operation of the bluff-body stabilized burner. In the present study, an electrical capacitance tomography (ECT) system was deployed to detect the permittivity distribution in the premixing channel and further characterize the flame states of stabilization, flashback, liftoff, extinction and blowout. A calderon-based reconstruction method was modified to reconstruct the permittivity distribution in the annular premixing channel. The detection results indicate that the permittivity in the premixing channel increases steeply when the flame flashback takes place and decreases obviously when the flame lifts off from the combustor rim. Based on the varied permittivity distribution at different flame states, a flame state index was proposed to characterize the flame state in quantification. The flame state index is 0, positive, in the range of −0.64–0, and lower than −0.64 when the flame is at the state of stable, flashback, liftoff and blowout, respectively. The flame state index at the flame state of extinction is the same as that at the flame state of liftoff. The extinction state and the blowout state can be distinguished by judging whether the flame flashback takes place before the flame is extinguished. These results reveal that the ECT system is capable of monitoring the flame state, and that the proposed flame state index can be used to characterize the flame state.

The Impacts of Scavenging on the Prey-Predator-Scavenger Fishery Model in the Existence of Harvesting and Toxin

The current paper investigates a prey-predator-scavenger fishery model in which both prey and predator species release harmful toxins during predation interaction. The harvesting of scavengers is introduced due to their economic value and serves as one of the important protein sources for humans. Both predator and scavenger fish species consume prey as their food. Scavengers devour the carrion or carcasses of predators which die of natural causes and are infected by the toxins of their preys. The non-negativity of solutions from the fishery model has been derived to ensure biologically meaningful. The model's equilibria are investigated along with its local stability characteristics. We investigate the threshold conditions triggering the bifurcations to occur in the steady-states concerning the scavenging activities by the scavengers. By formulating a suitable Lyapunov function, the global stability analysis of the non-trivial or coexistence steady-state is implemented. The dynamical behaviours of the fishery model, as well as the persistence and extirpation properties, have been analysed using the bifurcation analysis. From the results, it is found that the scavenging parameter in the fishery model with the presence of harvesting and toxin can drive the fish population towards extinction state which is unstable and unfavourable in nature.

A coupled algebraic-delay differential system modeling tick-host interactive behavioural dynamics and multi-stability

We propose a coupled system of delay-algebraic equations to describe tick attaching and host grooming behaviors in the tick-host interface, and use the model to understand how this tick-host interaction impacts the tick population dynamics. We consider two critical state variables, the loads of feeding ticks on host and the engorged ticks on the ground for ticks in a particular development stage (nymphal stage) and show that the model as a coupled system of delay differential equation and an algebraic (integral) equation may have rich structures of equilibrium states, leading to multi-stability. We perform asymptotic analyses and use the implicit function theorem to characterize the stability of these equilibrium states, and show that bi-stability and quadri-stability occur naturally in several combinations of tick attaching and host grooming behaviours. In particular, we show that in the case when host grooming is triggered by the tick biting, the system will have three stable equilibrium states including the extinction state, and two unstable equilibrium states. In addition, the two nontrivial stable equilibrium states correspond to a low attachment rate and a large number of feeding ticks, and a high attachment rate and a small number of feeding ticks, respectively.

Sex-biased predation and predator intraspecific competition effects in a prey mating system

&lt;abstract&gt;&lt;p&gt;In this work, we propose and investigate a predator-prey model where the prey population is structured by sex and the predators (unstructured) depredate based on sex-bias. We provide conditions for the existence of equilibrium points and perform local stability analysis on them. We derive global stability conditions for the extinction state. We show the possible occurrence of Hopf and saddle-node bifurcations. Multiple Hopf bifurcations are observed as the sex-biased predation rate is varied. This variation also shows the opposite consequences in the densities of the sex-structured prey. Our results show that sex-biased predation can cause both stabilizing and destabilizing effects for certain parameter choices. It can also cause an imbalanced sex-ratio, which has ecological consequences. Furthermore when intraspecific competition among predators is minimized, it can lead to the extinction of prey. We discuss the ecological implications and application of our results to the biocontrol of invasive species susceptible to sex-biased predation.&lt;/p&gt;&lt;/abstract&gt;

Most probable trajectory of a tumor model with immune response subjected to asymmetric Lévy noise

This paper is devoted to making realistic prediction for the trajectory transition of tumor cells induced by stochastic disturbances. To maintain better consistency with reality, an asymmetric Lévy noise is incorporated in a tumor–immune model. The corresponding Fokker–Planck equation is derived by the adjoint operator method. The transition behaviors of tumor cells are measured by maximizing the probability density function of system trajectories, namely, the most probable trajectory. The numerical algorithm for obtaining the most probable trajectory is offered. Based on the numerical calculations, the effect of Lévy noise on the evolution trajectories of tumor cells is discussed to explore the optimal parameters controlling or even eradicating tumor. It is found that by adjusting noise parameters, Lévy noise can induce the transition of tumor from the high concentration stable state toward the low concentration stable state or even to the tumor extinction state. When the skewness parameter is negative, Lévy noise with the stability index less than 1.0 and larger noise intensity is in favor of inhibiting tumor proliferation. While when the skewness parameter is positive, Lévy noise with the stability index larger than 1.0 and smaller noise intensity promotes the decrease of tumor cells.

Stochastic differential equation models for tumor population growth

In this paper we develop stochastic differential equation models for tumor population growth with immunization. We investigate especially the effect of a multiplicative noise on the progression of tumor evolution. Then, we perform a computational analysis in order to assess the behavior of the solution process. With this, we analyze the population extinction probability and the probability of persistence. In the case of persistence we analyze the behavior of the steady probability density and the evolution of its mean and variance. We further calculate the mean first passage time in order to describe the probability transitions between the extinction state and the state of the stable tumor. It is found that an increasing noise intensity rate enhances the probability of the extinction of the tumor.

Dynamics and stability of two predators–one prey mathematical model with fading memory in one predator

This paper concentrates on dynamics and stability analysis of two predators–one prey mathematical model with competition between predators and fading memory in one predator. The investigation of the constructed model shows that there exist five equilibria, e.g. trivial extinction state of all populations, extinction of both predators state, extinction of first or second predator state and coexisting state. Investigating the eigenvalues of characteristic polynomial, conditions for the local stability around each equilibrium are also determined depending on the parameter space. Analytical formulations are complemented with numerical simulations, where time simulations and single parameter numerical continuation of each variable are performed with respect to model parameters and multiple sub-and super-critical Hopf bifurcations, period doubling bifurcation and transcritical bifurcation are detected for different values of memory related parameter. Our results show that fading memory and competition between predators have substantial impact on the existence and dynamics of all three populations and may shed lights on further understanding of interacting species in ecology.

Forced waves of a three species predator-prey system in a shifting environment

This paper is concerned with the existence and non-existence of forced waves for a three species predator-prey system in a shifting environment. The speed of these forced waves is the same as the shifting speed of the living environment of these species. We assume that their habitat changes to the hostile environment as time increases. This makes all species go extinction eventually. Under certain conditions on parameters, we obtain two different types (front and mixed front-pulse types) forced waves that connecting different constant states and the extinction state. Moreover, we are able to characterize the minimal shifting speed for each mixed front-pulse type forced waves.

Strong Allee effect and basins of attraction in a discrete‐time zoonotic infectious disease model

AbstractMotivated by the Feline immunodeficiency virus, the virus that causes AIDS in cat populations, we use discrete‐time infectious disease models with demographic strong Allee effect to examine the impact of the fatal susceptible‐infected (SI) infections on two different types of growth functions: Holling type III or modified Beverton–Holt per‐capita growth function (compensatory density dependence), and Ricker per‐capita growth function with mating (overcompensatory density dependence). The occurrence of the strong Allee effect in the disease‐free equation renders the SI population model bistable, where the two coexisting locally asymptotically stable equilibrium points are either the origin (catastrophic extinction state) and the second fixed point (compensatory dynamics) or the origin and an intrinsically generated demographic period population cycle (overcompensatory dynamics). We use the basic reproduction number, , and the spectral radius, , to examine the structures of the coexisting attractors. In particular, we use MATLAB simulations to show that the fatal disease is not only capable of enlarging or shrinking the basin of attraction of the catastrophic extinction state, but it is also capable of fracturing the basins of attraction into several disjoint sets. Thus, making it difficult to specify the asymptotic zoonotic SI disease outcome in terms of all initial infections. The complexity of the basins of attractions appears to increase with an increase in the period of the intrinsically generated demographic population cycles.Recommendations for Resource Managers: Worldwide, disease agents which originate from animals or products of animal origin, such as Ebola virus, COVID‐19, West Nile virus, or malaria are responsible for millions of deaths each year. To help improve control and ultimately eradicate future pandemics in human populations, we use a Feline immunodeficiency virus disease model with the strong Allee effect to understand outbreaks of zoonotic diseases. In general, the strong Allee effect renders a population bistable with a catastrophic extinction state coexisting with a persistence state. We use the basic reproduction number and the spectral radius to obtain conditions for outbreaks of zoonotic diseases and conditions for zoonotic disease extinction. We show that a fatal zoonotic disease is not only capable of enlarging or shrinking the basin of attraction of the catastrophic extinction state, but it is also capable of fracturing the basins of attraction into several disjoint sets. Thus, making it difficult to specify the asymptotic zoonotic disease outcome in terms of all initial infections. The complexity of the basins of attractions appears to increase with an increase in the period of the demographic population cycles.

Traveling waves for reaction-diffusion PDE coupled to difference equation with nonlocal dispersal term and time delay

We consider a class of biological models represented by a system composed of reactiondiffusion PDE coupled with difference equations (renewal equations) in n-dimensional space, with nonlocal dispersal terms and implicit time delays. The difference equation generally arises, by means of the method of characteristics, from an age-structured partial differential system. Using upper and lower solutions, we study the existence of monotonic planar traveling wave fronts connecting the extinction state to the uniform positive state. The corresponding minimum wave speed is also obtained. In addition, we investigate the effect of the parameters on this minimum wave speed and we give a detailed analysis of its asymptotic behavior.

Dynamics in a Predator–Prey Model with Cooperative Hunting and Allee Effect

This paper deals with a diffusive predator–prey model with two delays. First, we consider the local bifurcation and global dynamical behavior of the kinetic system, which is a predator–prey model with cooperative hunting and Allee effect. For the model with weak cooperation, we prove the existence of limit cycle, and a loop of heteroclinic orbits connecting two equilibria at a threshold of conversion rate p=p#, by investigating stable and unstable manifolds of saddles. When p&gt;p#, both species go extinct, and when p&lt;p#, there is a separatrix. The species with initial population above the separatrix finally become extinct, and the species with initial population below it can be coexisting, oscillating sustainably, or surviving of the prey only. In the case with strong cooperation, we exhibit the complex dynamics of system, including limit cycle, loop of heteroclinic orbits among three equilibria, and homoclinic cycle with the aid of theoretical analysis or numerical simulation. There may be three stable states coexisting: extinction state, coexistence or sustained oscillation, and the survival of the prey only, and the attraction basin of each state is obtained in the phase plane. Moreover, we find diffusion may induce Turing instability and Turing–Hopf bifurcation, leaving the system with spatially inhomogeneous distribution of the species, coexistence of two different spatial-temporal oscillations. Finally, we consider Hopf and double Hopf bifurcations of the diffusive system induced by two delays: mature delay of the prey and gestation delay of the predator. Normal form analysis indicates that two spatially homogeneous periodic oscillations may coexist by increasing both delays.