Coexistence Steady State Research Articles

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.

Global boundedness and dynamics of a diffusive predator–prey model with modified Leslie–Gower functional response and density-dependent motion

This paper is devoted to studying the dynamical behaviors and stationary patterns of a diffusive modified Leslie–Gower predator–prey model with density-dependent motion in the predator population. We establish the existence of classical solutions with the uniform-in time bound and then analyze the local and global stability of the spatially homogeneous co-existence steady state under certain parametric conditions. By choosing the prey diffusion rate d2 as the bifurcation parameter, the steady state bifurcations from the positive constant equilibrium solution are investigated. Numerical simulations are performed to corroborate our analytical findings.

Complex dynamics near extinction in a predator-prey model with ratio dependence and Holling type III functional response

In this paper, we analyze a recently proposed predator-prey model with ratio dependence and Holling type III functional response, with particular emphasis on the dynamics close to extinction. By using Briot-Bouquet transformation we transform the model into a system, where the extinction steady state is represented by up to three distinct steady states, whose existence is determined by the values of appropriate Lambert W functions. We investigate how stability of extinction and coexistence steady states is affected by the rate of predation, predator fecundity, and the parameter characterizing the strength of functional response. The results suggest that the extinction steady state can be stable for sufficiently high predation rate and for sufficiently small predator fecundity. Moreover, in certain parameter regimes, a stable extinction steady state can coexist with a stable prey-only equilibrium or with a stable coexistence equilibrium, and it is rather the initial conditions that determine whether prey and predator populations will be maintained at some steady level, or both of them will become extinct. Another possibility is for coexistence steady state to be unstable, in which case sustained periodic oscillations around it are observed. Numerical simulations are performed to illustrate the behavior for all dynamical regimes, and in each case a corresponding phase plane of the transformed system is presented to show a correspondence with stable and unstable extinction steady state.

Dynamics of a spatiotemporal model on populations in a polluted river

In this paper, we investigate the dynamics for a reaction–diffusion–advection system which models populations in a polluted river. More precisely, we study the stability of steady states, which yields sufficient conditions that lead to population persistence or extinction. Furthermore, some dependence of the stability of the toxicant-only steady state and the population-toxicant coexistence steady state on the model parameters are given.

Global dynamics of a two-species Lotka-Volterra competition-diffusion-advection system with general carrying capacities and intrinsic growth rates II: Different diffusion and advection rates

We study a two-species competition-diffusion-advection system with general intrinsic growth rates and carrying capacities in heterogeneous closed environments, where the two aquatic species have different advection and/or diffusion rates. Some special cases have been widely investigated, for example: the non-advective case [8,9], the homogeneous case [23,24,40], the proportional case [20,25]. However, due to the difficulty caused by the diffusion-advection type operators as well as the heterogeneous environments, the approaches developed in the previous papers cannot be directly applied to the general case here. With some new techniques, we characterize the linear stability of the two semi-trivial steady states and establish the non-existence of co-existence steady states in several scenarios. Based on these preparations and the theory of monotone dynamical systems, we establish the main results. Our results indicate that both competitive exclusion and coexistence may occur. Specially, we show a new way for coexistence of two species.

Structural sensitivity in the functional responses of predator–prey models

In mathematical modelling, several different functional forms can often be used to fit a data set equally well, especially if the data is sparse. In such cases, these mathematically different but similar looking functional forms are typically considered interchangeable. Recent work, however, shows that similar functional responses may nonetheless result in significantly different bifurcation points for the Rosenzweig–MacArthur predator–prey system. Since the bifurcation behaviours include destabilizing oscillations, predicting the occurrence of such behaviours is clearly important. Ecologically, different bifurcation behaviours mean that different predictions may be obtained from the models. These predictions can range from stable coexistence to the extinction of both species, so obtaining more accurate predictions is also clearly important for conservationists. Mathematically, this difference in bifurcation structure given similar functional responses is called structural sensitivity. We extend the existing work to find that the Leslie–Gower–May predator–prey system is also structurally sensitive to the functional response. Using the Rosenzweig–MacArthur and Leslie–Gower–May models, we then aim to determine if there is some way to obtain a functional description of data so that different functional responses yield the same bifurcation structure, i.e., we aim to describe data such that our model is not structurally sensitive. We first add stochasticity to the functional responses and find that better similarity of the resulting bifurcation structures is achieved. Then, we analyse the functional responses using two different methods to determine which part of each function contributes most to the observed bifurcation behaviour. We find that prey densities around the coexistence steady state are most important in defining the functional response. Lastly, we propose a procedure for ecologists and mathematical modellers to increase the accuracy of model predictions in predator–prey systems.

Global dynamics of a three-species spatial food chain model

In this paper, we study the following initial-boundary value problem of a three-species spatial food chain model{ut=d1Δu+u(1−u)−b1uv,x∈Ω,t>0vt=d2Δv−∇⋅(ξv∇u)+uv−b2vw−θ1v,x∈Ω,t>0wt=Δw−∇⋅(χw∇v)+vw−θ2w,x∈Ω,t>0 in a bounded domain Ω⊂R2 with smooth boundary and homogeneous Neumann boundary conditions, where all parameters are positive constants. By the delicate coupling energy estimates, we first establish the global existence of classical solutions in two dimensional spaces for appropriate initial data. Moreover by constructing Lyapunov functionals and using LaSalle's invariance principle, we establish the global stability of the prey-only steady state, semi-coexistence and coexistence steady states.

Dynamics of Oxygen-Plankton Model with Variable Zooplankton Search Rate in Deterministic and Fluctuating Environments

It is estimated by scientists that 50–80% of the oxygen production on the planet comes from the oceans due to the photosynthetic activity of phytoplankton. Some of this production is consumed by both phytoplankton and zooplankton for cellular respiration. In this article, we have analyzed the dynamics of the oxygen-plankton model with a modified Holling type II functional response, based on the premise that zooplankton has a variable search rate, rather than constant, which is ecologically meaningful. The positivity and uniform boundedness of the studied system prove that the model is well-behaved. The feasibility conditions and stability criteria of each equilibrium point are discussed. Next, the occurrence of local bifurcations are exhibited taking each of the vital system parameters as a bifurcation parameter. Numerical simulations are illustrated to verify the analytical outcomes. Our findings show that (i) the system dynamics change abruptly for a low oxygen production rate, resulting in depletion of oxygen and plankton extinction; (ii) the proposed system has oscillatory behavior in an intermediate range of oxygen production rates; (iii) it always has a stable coexistence steady state for a high oxygen production rate, which is dissimilar to the outcome of the model of a coupled oxygen-plankton dynamics where zooplankton consumes phytoplankton with classical Holling type II functional response. Lastly, the effect of environmental stochasticity is studied numerically, corresponding to our proposed system.

Bifurcation and chaos in a smooth 3D dynamical system extended from Nosé-Hoover oscillator

The chaotic system with complicated dynamical behaviors has high potential in practical applications. This paper reports a simple smooth 3D dynamical system that is derived from the Nosé-Hoover oscillator and has rich dynamical behaviors, such as fold-Hopf bifurcation, saddle-node bifurcation, transient chaos, and conservative chaos (hidden attractors), which respectively correspond to four cases of equilibrium: infinitely many equilibria, two equilibria, four equilibria and no equilibrium. We first concentrate on the theoretical investigation of degenerate fold-Hopf and saddle-node bifurcations. Then, the power-law distribution of average lifetime of transient chaos is calculated and the coexistence of steady state and periodic motion after transient chaos is partially exhibited by time series. For the case of no equilibrium, we further study the conservative and ergodic dynamics, and hidden attractors with extremely rare properties by time-domain waveforms, phase portraits, histograms and Poincaré sections. Finally, the FPGA-based experimental results are presented and they are consistent with the numerical simulation results.

Dynamics of an immune-epidemiological model with virus evolution and superinfection

To get a better understanding how virus evolution influences on the disease spread, in this paper, we develop a nested immune-epidemiological model with superinfection. The interaction between virokins of within exposed host is described by an ordinary differential equation model. For this within host model, besides the infect-free equilibrium, there are two boundary equilibria and a unique coexistence equilibrium. Virokins coexistence within exposed host leads to the superinfection occurs between hosts. To consider the superinfection between host, we proposed an age structured epidemic model with latency age, in which the transfer rate and the superinfection rate of the exposed individuals both depend on the latency age. For this between host model, the local stabilities of the two boundary steady states and the existence of coexistence steady state are totally determined by the invasion reproduction numbers, R12, R21. Especially, uniform persistence of the between host model is strictly proved by use of integral semigroup theory when R12>1 and R21>1. Some numerical simulations are carried out to verify the main theoretical results and to illustrate the two concerns ((i) the effect of virokins coexistence within exposed hosts on the superinfection between hosts, (ii) the influence of the virus evolution on the dynamics of the age structured model). Finally, a brief conclusion is given in Section 6.

Hopf bifurcation and Bautin bifurcation in a prey–predator model with prey’s fear cost and variable predator search speed

This paper considers an extended predator–prey model, in which the fear of predators reduces prey reproduction and the search speed of predators depends on prey density. First, our mathematical analysis shows that high levels of fear can stabilize the coexistence steady state, while low levels would result in periodic oscillation. Comparing to the model ignoring fear where supercritical bifurcation occurs, Hopf bifurcation in our model can be both supercritical and subcritical, which leads to bi-stability and two limit cycles. Second, our analysis demonstrates that a relatively small search speed of predators can promote stability of the coexistence steady state, while a large speed would lead to periodic oscillation. Comparing to the model with invariant search speed where Hopf bifurcation takes place, Bautin bifurcation occurs in our model, which results in tri-stability and three limit cycles. While the paradox of enrichment always takes place in the Holling-type II predation model, it does not occur here when the search speed is small. Even when the speed is large, the prey species can adapt by enhancing their fear level and stabilize the system effectively. Third, our analysis shows that enhancing prey’s sensitivity to predation risk or slowing the predator search speed, can stabilize the coexistence steady state, while a low sensitivity and a high speed will lead to periodic oscillation. Numerical simulations confirm and extend our results.

Spatiotemporal pattern formation in a prey–predator model with generalist predator

Generalist predators exploit multiple food sources and it is economical for them to reduce predation pressure on a particular prey species when their density level becomes comparatively less. As a result, a prey-predator system tends to become more stable in the presence of a generalist predator. In this article, we investigate the roles of both the diffusion and nonlocal prey consumption in shaping the population distributions for interacting generalist predator and its focal prey species. In this regard, we first derive the conditions associated with Turing instability through linear analysis. Then, we perform a weakly nonlinear analysis and derive a cubic Stuart-Landau equation governing amplitude of the resulting patterns near Turing bifurcation boundary. Further, we present a wide variety of numerical simulations to corroborate our analytical findings as well as to illustrate some other complex spatiotemporal dynamics. Interestingly, our study reveals the existence of traveling wave solutions connecting two spatially homogeneous coexistence steady states in Turing domain under the influence of temporal bistability phenomenon. Also, our investigation shows that nonlocal prey consumption acts as a stabilizing force for the system dynamics.

Global dynamics of a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and prey-taxis

<abstract><p>In this paper, our purpose is to discuss the global dynamics of a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and prey-taxis under homogeneous Neumann boundary conditions. First, we derive that the global classical solutions of the system are globally bounded by taking advantage of the Morse's iteration of the parabolic equation, which further arrives at the global existence of classical solutions with a uniform-in-time bound. In addition, we establish the global stability of the spatially homogeneous coexistence steady states under certain conditions on parameters by constructing Lyapunov functionals.</p></abstract>

Reaction-advection-diffusion competition models under lethal boundary conditions

<p style='text-indent:20px;'>In this study, we consider a Lotka–Volterra reaction–diffusion–advection model for two competing species under homogeneous Dirichlet boundary conditions, describing a hostile environment at the boundary. In particular, we deal with the case in which one species diffuses at a constant rate, whereas the other species has a constant rate diffusion rate with a directed movement toward a better habitat in a heterogeneous environment with a lethal boundary. By analyzing linearized eigenvalue problems from the system, we conclude that the species dispersion in the advection direction is not always beneficial, and survival may be determined by the convexity of the environment. Further, we obtain the coexistence of steady-states to the system under the instability conditions of two semi-trivial solutions and the uniqueness of the coexistence steady states, implying the global asymptotic stability of the positive steady-state.</p>

Dynamic analysis of a new aquatic ecological model based on physical and ecological integrated control.

Within the framework of physical and ecological integrated control of cyanobacteria bloom, because the outbreak of cyanobacteria bloom can form cyanobacteria clustering phenomenon, so a new aquatic ecological model with clustering behavior is proposed to describe the dynamic relationship between cyanobacteria and potential grazers. The biggest advantage of the model is that it depicts physical spraying treatment technology into the existence pattern of cyanobacteria, then integrates the physical and ecological integrated control with the aggregation of cyanobacteria. Mathematical theory works mainly investigate some key threshold conditions to induce Transcritical bifurcation and Hopf bifurcation of the model (2.1), which can force cyanobacteria and potential grazers to form steady-state coexistence mode and periodic oscillation coexistence mode respectively. Numerical simulation works not only explore the influence of clustering on the dynamic relationship between cyanobacteria and potential grazers, but also dynamically show the evolution process of Transcritical bifurcation and Hopf bifurcation, which can be clearly seen that the density of cyanobacteria decreases gradually with the evolution of bifurcation dynamics. Furthermore, it should be worth explaining that the most important role of physical spraying treatment technology can break up clumps of cyanobacteria in the process of controlling cyanobacteria bloom, but cannot change the dynamic essential characteristics of cyanobacteria and potential grazers represented by the model (2.1), this result implies that the physical spraying treatment technology cannot fundamentally eliminate cyanobacteria bloom. In a word, it is hoped that the results of this paper can provide some theoretical support for the physical and ecological integrated control of cyanobacteria bloom.

Repulsive chemotaxis and predator evasion in predator–prey models with diffusion and prey-taxis

The role of predator evasion mediated by chemical signaling is studied in a diffusive prey–predator model when prey-taxis is taken into account (model A) or not (model B) with taxis strength coefficients [Formula: see text] and [Formula: see text], respectively. In the kinetic part of the models, it is assumed that the rate of prey consumption includes functional responses of Holling, Beddington–DeAngelis or Crowley–Martin. Existence of global-in-time classical solutions to model A is proved in space dimension [Formula: see text] while to model B for any [Formula: see text]. The Crowley–Martin response combined with bounded rate of signal production precludes blow-up of solution in model A for [Formula: see text]. Local and global stability of a constant coexistence steady state which is stable for the corresponding ordinary differential equation (ODE) and purely diffusive model are studied along with mechanism of Hopf bifurcation for model B when [Formula: see text] exceeds some critical value. In model A, it is shown that prey-taxis may destabilize the coexistence steady state provided [Formula: see text] and [Formula: see text] are big enough. Numerical simulation depicts emergence of complex space-time patterns for both models and indicates existence of solutions to model A which blow-up in finite time for [Formula: see text].

Indirect taxis drives spatio-temporal patterns in an extended Schoener’s intraguild predator–prey model

In Schoener’s model of intraguild-predation a prey–predator interaction is mixed with the competition of the prey and the predator for food resource supplied to a system with a constant rate. In this work the model is extended to examine the impact of indirect prey taxis which counts for the movement of predator towards the odor released by prey rather than directly towards gradient of prey density (prey taxis) and indirect predator taxis which refers to prey movement opposite to the gradient of a chemical released by predator. The constant coexistence steady state in the model was shown earlier to be globally stable when Schoener’s O.D.E. model is generalized to reaction–diffusion or even prey taxis system. Existence of global-in-time solutions to Schoener’s model with indirect prey taxis is proved for any space dimension while for the case of indirect predator taxis only in 1D. This study reveals that sufficiently large value of taxis sensitivity parameter disturbs the stability of the coexistence steady state giving rise to pattern formation governed by the Hopf bifurcation. Numerical simulations illustrate emergence of taxis driven spatio-temporal periodic patterns.

Impact of Fear and Habitat Complexity in a Predator-Prey System with Two Different Shaped Functional Responses: A Comparative Study

Habitat complexity or the structural complexity of habitat reduces the available space for interacting species, and subsequently, the encounter rate between the prey and predator is decreased significantly. Different experimental shreds of evidence validate that the presence of the predator strongly affects the physiological behaviour of prey individuals and dramatically reduces their reproduction rate. In this study, we investigate the interplay between the level of fear and the degree of habitat complexity in a predator-prey model with two different shaped functional responses. We, therefore, develop the functional response using the timescale separation method, and the shape of the resulting functional response depends upon the monotonous property of catch rate, g N where N is the prey biomass. Whenever g N increases strictly, a saturating functional response occurs, but for nonmonotonic g N , a dome-shaped functional response arises. For saturating case, it has been revealed that both prey and predator biomass may oscillate for lower levels of fear and a lower degree of habitat complexity. To stabilize this oscillatory behaviour to a coexistence state, we have to adequately increase the level of fear or degree of habitat complexity. However, for dome-shaped case, more complicated dynamics are observed. In this case, coexistence steady state, if exists, may be locally asymptotically stable for a lower degree of habitat complexity, but for intermediate values, the system is capable of producing multiple coexistence steady states with a bistable phenomenon between predator-free steady state and a coexistence steady state. Moreover, if the level of fear is sufficiently low, the system may experience a supercritical or/and subcritical Hopf bifurcation. In the dynamics of parametric disturbance for the degree of habitat complexity parameter, dome-shaped functional response predicts that disturbance may trap the system into a nearest attractor (either a large amplitude stable limit cycle or predator-free steady state); this can be overcome only by a larger alteration, or sometimes it is impossible to overcome (hysteresis phenomena), whereas the saturating-shaped functional response predicts a system resilience. For both the functional responses, a higher degree of habitat complexity always increases the extinction possibility of the predator, and no level of fear can compensate this biodiversity loss.

Bifurcation structure of coexistence states for a prey–predator model with large population flux by attractive transition

This paper is concerned with a prey–predator model with population flux by attractive transition. Our previous paper (Oeda and Kuto, 2018, Nonlinear Anal. RWA, 44, 589–615) obtained a bifurcation branch (connected set) of coexistence steady states which connects two semitrivial solutions. In Oeda and Kuto (2018, Nonlinear Anal. RWA, 44, 589–615), we also showed that any positive steady-state approaches a positive solution of either of two limiting systems, and moreover, one of the limiting systems is an equal diffusive competition model. This paper obtains the bifurcation structure of positive solutions to the other limiting system. Moreover, this paper implies that the global bifurcation branch of coexistence states consists of two parts, one of which is a simple curve running in a tubular domain near the set of positive solutions to the equal diffusive competition model, the other of which is a connected set characterized by positive solutions to the other limiting system.

Impact of fear in a delay-induced predator–prey system with intraspecific competition within predator species

In theoretical ecology, predator–prey interaction is a natural phenomenon that significantly contributes for shaping the community structure and maintaining the ecological diversity. In almost every ecological model, the prey species is curtailed by the direct attacking of predator species. However, from different experimental shreds of evidence, it has been observed that fear (felt by prey) for predators can change the physiological behaviour of prey individuals and greatly reduces their reproduction rate as well as enhances their mortality rate. In this current work, we develop and explore a predator–prey model incorporating the cost of perceived fear into the birth and death rates of prey species with Holling type-II functional response. In addition, the intraspecific competition within predator species and a gestation delay are introduced in the model to obtain more realistic and natural dynamics. Feasibility of all the steady states and their stability conditions are analysed in terms of the model parameters. We show that only existence of an interior equilibrium point is sufficient to prevent the extinction of predator species. In this case, either both species can exist together or oscillate around that interior equilibrium point. We can also recognize the parametric region where the system produces multiple coexistence equilibria in which different initial biomass of populations may produce different long-term outcomes. The basic bifurcation analyses of the system exhibit that a higher level of fear or higher intraspecific competition rate helps the population to survive in a coexistence state. For a suitable choice of parametric values, the proposed model may produce the bi-stable phenomenon between two coexistence steady states. We obtain a parametric condition for which the model system experiences a Hopf bifurcation if the delay parameter exceeds some threshold value. All of these theoretical findings are verified by various numerical examples.