Coexistence Steady State Research Articles

Global boundedness and asymptotic stability for a food chain model with nonlinear diffusion

This paper is concerned with a food chain model with nonlinear diffusion ut = Δu + u(1 − u − b1v), vt=∇⋅((v+1)m∇v)−∇⋅(ξv∇u)+vu−b2w1+v+w−θ1−α1v,wt=∇⋅((w+1)l∇w)−∇⋅(χw∇v)+wv1+v+w−θ2−α2w in a smooth bounded domain Ω ⊂ Rn(n ≥ 2) with homogeneous Neumann boundary conditions, where the parameters ξ, χ, α1, bi, θi (i = 1, 2) > 0 and α2 ≥ 0 as well as m, l∈R. We study the global boundedness of classical solutions to the problem if either n = 2 and m ≥ 0, l > − 1 or n ≥ 3 and m>1−2n, l > − 1. Moreover, we prove the global stability of the prey-only steady state and semi-coexistence steady as well as coexistence steady states under certain conditions on parameters.

Analysis of an intra- and interspecific interference model with allelopathic competition

In this paper, we propose an original model of two-microbial species competing for a single nutrient in the chemostat including general intra- and interspecific density-dependent growth rates with allelopathic interactions. Each species produces a toxin that affects the growth of other species as well as its own growth. The removal rates are distinct and include the specific death rate and the autotoxicity of each species. We establish an in-depth mathematical analysis by determining the multiplicity of all steady states of the three-dimensional system and their necessary and sufficient conditions of existence and local stability according to the operating parameters, which are the dilution rate and the inflowing concentration of the substrate. To describe the asymptotic behavior of the process according to these control parameters, we first determine theoretically the operating diagram. Using MATCONT software, these theoretical results are validated numerically but it reveals the cusp bifurcation that occurs by varying two parameters. The one-parameter bifurcation diagram in the dilution rate shows that there can be either transcritical or saddle-node bifurcations. Finally, we apply our results to a particular model in the literature without intra- and interspecific interference but with only allelopathic effects of the second species on the first species. We show that one of the coexistence steady states is locally exponentially stable when it exists, whereas in the literature they have not been able to demonstrate that the stability condition of this steady state is always fulfilled.

Coexistence of two strongly competitive species in a reaction–advection–diffusion system

The main focus of this article is to investigate the behavior of two strongly competitive species in a spatially heterogeneous environment using a Lotka–Volterra-type reaction–advection–diffusion model. The model assumes that one species diffuses at a constant rate, while the other species moves toward a more favorable environment through a combination of constant diffusion and directional movement. The study finds that no stable coexistence can be guaranteed when both species disperse randomly. In contrast, stable coexistence between the two species is possible when one of the species exhibits advection–diffusion. The study also reveals the existence of unstable coexistence imposed by bistability in a strongly competitive system, regardless of the diffusion type. The results are obtained by analyzing the stability of semitrivial solutions. The study concludes that the species moving toward a better environment has a competitive advantage, allowing them to survive even when their population density is initially low. Finally, the study identifies the unique globally asymptotically stable coexistence steady states of the system at high advection rates, particularly for relatively moderate interspecific competition parameters in species with directional movement. These findings underscore the crucial role of directed movement and interspecific competition coefficients in shaping the dynamics and coexistence of strongly competing species.

Dynamics of reaction–diffusion–advection system and its impact on river ecology in the presence of spatial heterogeneity

This study considers a two-species reaction–diffusion–advection (RDA) model in a heterogeneous advective environment with zero Neumann boundary conditions. We suppose that two species compete for the same food resource but have different diffusion and advection rates. The primary objective of this study is to investigate the global asymptotic stability and coexistence of steady state based on different and unequal diffusion and advection rates through theoretical and numerical analysis. We establish the local stability of two semi-trivial steady states of the competing species. Also, the non-existence of coexistence steady state is proved with the help of some non-trivial assumptions. Finally, we show the global stability with the help of monotone dynamical systems and combine the local stability and non-existence of coexistence steady state. If one species adopts a smaller diffusion and advection rate, the competition’s result depends on the advection and diffusion rate ratio. If the ratio is smaller, then the species will prevail. Also, the species with higher diffusion and smaller advection will win. Lastly, the effectiveness of the model in one and two-dimensional situations is shown by a series of numerical calculations, which is particularly important for environmental consideration.

Modeling and analyzing competitive epidemic diseases with partial and waning virus-specific and cross-immunity

In this paper, we consider a novel mathematical modeling framework for the spread of two competitive diseases in a well-mixed population. The proposed framework, which we term a bivirus SIRIS model, encapsulates key real-world features of natural immunity, accounting for different levels of (partial and waning) virus-specific and cross protection acquired after recovery. Formally, the proposed framework consists of a system of coupled nonlinear ordinary differential equations that builds on a classical bivirus susceptible–infected–susceptible model by means of the addition of further states to account for (temporarily) protected individuals. Through the analysis of the proposed framework and of two specializations, we offer analytical insight into how natural immunity can shape a wide range of complex emergent behaviors, including eradication of both diseases, survival of the fittest one, or even steady-state co-existence of the two diseases.

Global dynamics of a two-species clustering model with Lotka–Volterra competition

This paper is concerned with the global dynamics of a two-species Grindrod clustering model with Lotka–Volterra competition. The model takes the advective flux to depend directly upon local population densities without requiring intermediate signals like attractants or repellents to form the aggregation so as to increase the chances of survival of individuals like human populations forming small nucleated settlements. By imposing appropriate boundary conditions, we establish the global boundedness of solutions in two-dimensional bounded domains. Moreover, we prove the global stability of spatially homogeneous steady states under appropriate conditions on system parameters, and show that the rate of convergence to the coexistence steady state is exponential while the rate of convergence to the competitive exclusion steady state is algebraic.

Study of a cannibalistic prey–predator model with Allee effect in prey under the presence of diffusion

In this study, we have investigated the temporal and spatio-temporal behavior of a prey–predator model with weak Allee effect in prey and the quality of being cannibalistic in a specialist predator. The parameters responsible for the Allee effect and cannibalism impact both the existence and stability of coexistence steady states of the temporal system. The temporal system exhibits various kinds of local bifurcations such as saddle–node, Hopf, Generalized Hopf (Bautin), Bogdanov–Takens, and global bifurcation like homoclinic, saddle–node bifurcation of limit cycles. For the model with self-diffusion, we establish the non-negativity and prior bounds of the solution. Subsequently, we derive the theoretical conditions in which self-diffusion leads to the destabilization of the interior equilibrium. Additionally, we explore the conditions under which cross-diffusion induces the Turing-instability where self-diffusion fails to do so. Further, we present different kinds of stationary and dynamic patterns on varying the values of diffusion coefficients to depict the spatio-temporal model’s rich dynamics. It has been found that the addition of self and cross-diffusion in a prey–predator model with the Allee effect in prey and cannibalistic predator play essential roles in comprehending the pattern formation of a distributed population model. It is expected that the comprehensive mathematical analysis and extensive numerical simulations used in investigating the global dynamics of the proposed model can facilitate researchers in studying the temporal and spatial aspects of prey–predator models in more significant detail.

Time-delayed models for the effects of toxicants on populations in contaminated aquatic ecosystems

Ecotoxicological models play a vital role in understanding the influence of toxicants on population dynamics in contaminated aquatic ecosystems. Traditional differential equation models describing interactions between populations and toxicants typically assume instantaneous population growth, overlooking potential time delays associated with reproductive and maturation processes. In this paper, we introduce two models with time delays to investigate the interaction between a population and a toxicant, where the population growth is governed by a delayed logistic equation. We mainly focus on the stability analysis of the steady states of the models. Our findings indicate that high toxicant concentrations result in population extinction, whereas moderate toxicant levels can potentially induce bistability, where the population's fate, whether persistence or extinction, depends on the initial densities of the population and toxicant. Furthermore, both our theoretical analysis and numerical simulations demonstrate that the time delay can lead to the destabilization of the coexistence steady states and the appearance of periodic solutions through Hopf bifurcation.

Global dynamics of two-species reaction–diffusion competition model with Gompertz growth

ABSTRACT In this paper, we investigate a two-species reaction–diffusion competition model with Gompertz growth, where the intrinsic growth rates and carrying capacities of environments are heterogeneous. At firstly, assuming two competing species only admit different diffusive rates, we show that ‘slower diffuser prevails’, which is consistent with the well-known result in Dockery J, Hutson V, Mischaikow K, Pernarowski M. [The evolution of slow dispersal rates: a reaction–diffusion model. J Math Biol. 1998;37(1):61–83; Hastings A. Can spatial variation alone lead to selection for dispersal? Theor Popul Biol. 1983;24:244–251]. Then, for the “weak competition” case, we establish a prior estimate, which combined with the theory of monotone dynamical system and spectral analysis implies that the model admits a unique coexistence steady state, which is globally asymptotically stable. Finally, for the “strong–weak competition” case, we give the expression of critical competition intensity and the weak competitor will be wiped out.

Global dynamics and spatiotemporal heterogeneity of a preytaxis model with prey-induced acceleration

Abstract Conventional preytaxis systems assume that prey-tactic velocity is proportional to the prey density gradient. However, many experiments exploring the predator–prey interactions show that it is the predator’s acceleration instead of velocity that is proportional to the prey density gradient in the prey-tactic movement, which we call preytaxis with prey-induced acceleration. Mathematical models of preytaxis with prey-induced acceleration were proposed by Arditi et al. ((2001) Theor. Popul. Biol. 59(3), 207–221) and Sapoukhina et al. ((2003) Am. Nat. 162(1), 61–76) to interpret the spatial heterogeneity of predators and prey observed in experiments. This paper is devoted to exploring the qualitative behaviour of such preytaxis systems with prey-induced acceleration and establishing the global existence of classical solutions with uniform-in-time bounds in all spatial dimensions. Moreover, we prove the global stability of spatially homogeneous prey-only and coexistence steady states with decay rates under certain conditions on system parameters. For the parameters outside the stability regime, we perform linear stability analysis to find the possible patterning regimes and use numerical simulations to demonstrate that spatially inhomogeneous time-periodic patterns will typically arise from the preytaxis system with prey-induced acceleration. Noticing that conventional preytaxis systems are unable to produce spatial patterns, our results imply that the preytaxis with prey-induced acceleration is indeed more appropriate than conventional preytaxis to interpret the spatial heterogeneity resulting from predator–prey interactions.

Population dynamics in a reaction–diffusion-advection predator–prey model with Beddington–DeAngelis functional response

In this paper, we consider a two-species predator–prey model in advective heterogeneous environments with the Beddington–DeAngelis interaction term, where the Danckwerts boundary conditions are imposed. Applying the comparison principle for predator–prey system and persistence theory, we draw a clear picture on the long-time dynamics, which also indicates the existence and non-existence of the coexistence steady state solutions. Specially, the uniqueness of the coexistence steady state solutions is considered under certain conditions.

Symmetry in a multi-strain epidemiological model with distributed delay as a general cross-protection period and disease enhancement factor

Important biological features of viral infectious diseases caused by multiple agents with interacting strain dynamics continue to pose challenges for mathematical modelling development. Motivated by dengue fever epidemiology, we study a system of Integro-Differential Equations (IDE) considering strain structure of pathogens. Knowing that complex dynamics observed in dengue models are driven by the combination of two biological features, the temporary cross-immunity (TCI) and disease enhancement via the antibody-dependent enhancement process (ADE), our IDE system incorporates the TCI with a general time delay term, and the ADE effect by a constant factor to differentiate the susceptibility of individuals experiencing a primary or a secondary infection. Aiming at analysing the effect of the symmetry on dengue serotypes in the IDE framework, a detailed qualitative analysis of the model is performed and the instability of the coexistence steady state is shown using the perturbation theory approach. Numerical simulations identify the bifurcation structures and confirm the stability analysis. Results for the symmetric and asymmetric models are discussed.

Double-Hopf bifurcation and Pattern Formation of a Gause-Kolmogorov-Type system with indirect prey-taxis and direct predator-taxis

This paper focuses on a general Gause-Kolmogorov-Type predator–prey model with direct predator-taxis and indirect prey-taxis. The global existence and boundedness of solutions to the system are proved. Our approach is applicable to more general situation. Some measures of success for the constant coexistence steady state are rigorously calculated, such as decreasing the indirect prey-taxis sensitivity or the release rate of stimulus, and increasing the predator-taxis sensitivity or the decay rate of stimulus. By choosing the indirect prey-taxis coefficient as the bifurcation parameter, the existence of Hopf and double-Hopf bifurcations is established. We find that both indirect prey-taxis and direct predator-taxis can promote the complexity of the spatiotemporal pattern, e.g. spatially nonhomogeneous periodic patterns with different spatial frequency, spatially nonhomogeneous quasi-periodic patterns. Finally, we apply our theoretical analyses to a Rosenzweig–MacArthur model with indirect prey-taxis and direct predator-taxis, spatially homogeneous periodic patterns.

Global dynamics of a three-species spatial food chain model with alarm-taxis and logistic source

This work considers the following food chain model with alarm-taxis and logistic source ut=d1Δu+u(1−u)−b1uv,x∈Ω,t>0,vt=d2Δv−ξ∇⋅(v∇u)+uv−b2vw+θ1(v−v2),x∈Ω,t>0,wt=Δw−χ∇⋅(w∇(uv))+vw+θ2(w−w2),x∈Ω,t>0in a smoothly bounded domain Ω⊂RN(N≥3) with zero-flux boundary conditions, where the parameters di, bi, θi(i=1,2), ξ and χ are positive constants. If the positive parameters θ1 and θ2 are large enough, we show that this model possesses a globally bounded classical solution. Moreover, by constructing the Lyapunov functionals, we prove the global stability of the semi-coexistence steady states and the coexistence steady state under suitable conditions on parameters.

Mathematical analysis of a multiscale hepatitis C virus infection model with two viral strains

Direct-acting antiviral agents (DAAs) have shown higher cure rates for treating hepatitis C virus (HCV) by directly interfering with different steps of the HCV life cycle. To optimize drug combinations and accurately quantify the antiviral effect of DAAs treatments, mathematical models should include the intracellular virus replication processes. In this study, we develop a two-strain multiscale age-structured model that includes intracellular viral RNA replication and extracellular viral infection. We prove the existence, positivity, and boundedness of the solution, and derive the conditions for the existence and stability of the three steady states (non-infection steady state, boundary steady state, and coexistence steady state). Under certain biologically reasonable assumptions, we obtain the approximate solutions of the viral load decline of the two virus strains (sensitive and drug-resistant virus strains) during treatment, and the long-term approximation is shown to be in good agreement with the solution of the original model. We also carry out numerical simulations using the long-term approximate solution, providing insights into the influence of antiviral effects on the viral load change of the two virus strains during treatment with DAAs.

Mathematical analysis of the dynamics of some reaction-diffusion models for infectious diseases

We study some reaction-diffusion systems for infectious diseases and investigate how the dynamics is impacted by the movement of populations and spatial heterogeneity of the environment. General conditions for the existence, uniqueness and stability of the coexistence steady states are established. Our analysis revealed two mechanisms for the coexistence of strains: (i) when the ratio of the transmission probability of two strains falls into some intermediate ranges; (ii) when the diffusion rate of two strains falls into proper ranges. Interestingly, when there is no coexistence of strains, it is possible for the “weak” strain to be dominant for some intermediate range of diffusion rates, in strong contrast to small and large diffusion cases where the “weak” strain always goes extinct.

Consequences of Traceable Mobility in Populations Exhibiting Strong Allee Effect

In this research, we study the impacts of the traceable mobility in a two-patch environment when the population in each patch exhibits strong Allee effects. Traveling individuals are traced across patches by budgeting the average time spent in each patch while keeping their place of residency. Particularly, we focus on the impact that the effective population (residents and visitors) produces on regional dynamics.Our results show that low mobility across regions produces simple dynamics, where orbits converge to single or double extinction, or to a coexistence steady state. We derive mobility conditions under which an endangered population may benefit of the presence of a visitant one and avoid extinction -- the rescue effect. Nonetheless, increments in the visiting population would also lead the resident population to extinction -- the induced extinction effect.

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.

Maturation delay induced stability enhancement and shift of bifurcation thresholds in a predator–prey model with generalist predator

Discrete time delay has been widely used in models of interacting populations to capture various biological processes such as gestation, maturation, food digestion, disease transmission and so on. In most of the works, it has been demonstrated that a longer maturation delay is responsible for the destabilization of the coexistence steady-state. The constructed model system, however, may produce different phenomena if the maturation delay is included in an ecologically justified way, preventing the model from producing inconsistent results. Previous studies focus on the influence of delayed maturation in predator–prey dynamics with specialist predators mostly. The primary goal of this article is to explore how delayed maturity in generalist predators qualitatively affects the dynamics of a predator–prey system. To illustrate, we consider a predator–prey model with generalist predators and Holling type II functional response to characterize the generalist predator’s grazing behavior toward their primary prey. We find various parameter regimes for the non-delayed system where we see either no coexistence, single coexistence, or multiple coexistence of prey and their generalist predators. Rich dynamics, induced by various local and global bifurcations, are displayed for the non-delayed system. The inclusion of a maturation delay in the model system causes the system’s qualitative properties to alter, such as shifting bifurcation thresholds and a reduction in the region of multi-coexistence in a parametric domain. Our analysis demonstrates that the delayed maturation of generalist predators does not act as a destabilizing factor; instead, it promotes stable coexistence. Visual depictions of all the analytical findings have been presented through extensive numerical simulation and the ecological implications of the obtained results are summarized in the concluding section.

On the strongly competitive case in a fully parabolic two-species chemotaxis system with Lotka-Volterra competitive kinetics

This work considers a two-species chemotaxis system with Lotka-Volterra competitive kinetic functional response term in a bounded domain with smooth boundary. We proved global bounded solutions to the system in high dimensions (n≤5) without the convexity of the domain. Moreover, by constructing appropriate Lyapunov functionals, it is proved that the solution convergences to the semi-trivial steady state in L∞(Ω) under strong competition (a1,a2≥1, where a1,a2 represent the intensity of competition between different species) if the growth coefficients of two species are appropriately large. Compared to previous work, our result removes the requirement for the convexity of the domain and proves global bounded solutions in high dimensions (n≤5). Moreover, we studied the asymptotic behavior of solutions in the case of strong competition which is obscure in the existing literature. Furthermore, the linear stability analysis is performed to find the possible patterning regimes, outside the stability parameters regime, for both semi-trivial and coexistence steady states, our numerical simulations show that non-constant steady states and spatially inhomogeneous temporal periodic patterns are all possible.