Two-species Competition Model Research Articles

Oscillations and coexistence generated by discrete delays in a two-species competition model

Oscillations and coexistence generated by discrete delays in a two-species competition model

Global dynamics of a competitive system with seasonal succession and different harvesting strategies

In this paper, we investigate a two-species competition model with seasonal succession and different harvesting strategies. We aim to comprehend the impact of the proportion of good season on species competition outcomes and harvesting practices. For the case of weak harvest, we establish proportion thresholds for the good season for the species. Based on these threshold settings, we determine conditions for the existence and uniqueness of two semi-trivial periodic solutions and the existence, uniqueness and stability of positive periodic solutions under certain conditions. Numerical simulations and brief discussions are presented to validate our theoretical findings.

Propagation dynamics of two-species competition models in a periodic discrete habitat

This paper is devoted to the study of the propagation dynamics of a Lotka-Volterra two-species competition system in a periodic discrete habitat. Under appropriate assumptions, we show that a semi-trivial equilibrium is globally stable for the spatially periodic initial value problem. Then we establish the existence of the rightward spreading speed and its coincidence with the minimal wave speed for the spatially periodic rightward traveling waves. We further obtain sufficient conditions for the linear determinacy of the rightward spreading speed. Finally, we apply these results to a specific model of two-species competition and conduct numerical simulations for the spreading speed and traveling waves.

Numerical Investigation and Factor Analysis of Two-Species Spatial-Temporal Competition System after Catastrophic Events

The interaction of species in an ecological community can be described by coupled system partial differential equations. To analyze the problem numerically, we construct a discrete system using finite volume approximation by space with semi-implicit time approximation to decouple a system. We first simulate the converges of the system to the final equilibrium state for given parameters (reproductive rate, competition rate, and diffusion rate), boundaries, and initial conditions of population density. Then, we apply catastrophic events on a given geographic position with given catastrophic sizes to calculate the restoration time and final population densities for the system. After that, we investigate the impact of the parameters on the equilibrium population density and restoration time after catastrophe by gradually releasing the hold of different parameters. Finally, we generate data sets by solutions of a two-species competition model with random parameters and perform factor analysis to determine the main factors that affect the restoration time and final population density after catastrophic events.

Evolution of Dispersal in Advective Patchy Environments

We study a two-species competition model in a patchy advective environment, where the species are subject to both directional drift and undirectional random dispersal between patches and there are losses of individuals in the downstream end (e.g., due to the flow into a lake or ocean). The two competing species are assumed to have the same growth rates but different advection and random dispersal rates. We focus our studies on the properties of an associated eigenvalue problem which characterizes the extinction/persistence dynamics of the underlying patch population model. We also derive conditions on the advection and random dispersal rates under which a mutant species can or cannot invade the resident species.

Impact of correlated Gaussian colored noises on stability and stationary probability density for the randomly forced two-species competitive Gompertz model

In this paper, random factors described by correlated Gaussian colored noises are introduced into a two-species competition model with Gompertz growth. Firstly, stochastic stability is studied with the help of singular boundary theory. After that, the stationary probability density of the limiting system is given explicitly and then the extrema of invariant measure are discussed in detail. Finally, the average time of system escaping from the attraction domain is calculated using mean first passage time, and noise enhanced stability is observed. Our results show that random factors are more likely to make the system lose its original stability. Instead, the sample trajectories will stay near the maximum of the stationary probability density for a long time. Importantly, noise intensity and correlation degree are destructive, which can expand the distribution range of population density, weaken the stability of the system, and then easily lead to ecological imbalance. Conversely, the role of correlation time and proper competition between species is constructive and conducive to coexistence.

Population dynamics with resource-dependent dispersal: single- and two-species models.

In this paper, we consider the population models with resource-dependent dispersal for single-species and two-species with competition. For the single-species model, it is well-known that the total population supported by the environment is always greater than the environmental carrying capacity if the dispersal is simply random diffusion. However, we find that the total population supported can be equal or smaller than the environmental carrying capacity when the dispersal depends on the resource distribution. This analytical finding not only well agrees with the yeast experiment observation of Zhang et al. (Ecol Lett 20(9):1118-1128, 2017), but also indicates that resource-dependent dispersal may produce different outcomes compared to the random dispersal. For the two-species competition model, when two competing species use the same dispersal strategy up to a multiplicative constant (i.e. their dispersal strategies are proportional) or both diffusion coefficients are large, we give a classification of global dynamics. We also show, along with numerical simulations, that if the dispersal strategies are resource-dependent, even one species has slower diffusion, coexistence is possible though competitive exclusion may occur under different conditions. This is distinct from the prominent result that with random dispersal the slower diffuser always wipes out its fast competitor. Our analytical results, supported by the numerical simulations, show that the resource-dependent dispersal strategy has profound impact on the population dynamics and evolutionary processes.

Global dynamics of a two-species competition model in advective homogeneous environments

We investigate a two-species competition model in advective homogeneous environments. We assume that the two species are identical except their diffusion rates and advection rates. It is worth noting that the Neumann boundary condition is assumed at upstream end, meanwhile, there is always a net loss of individuals incurred by water flow or both water flow and diffusive movements at downstream end. Our results show that when the downstream loss is large (incurred by both water flow and diffusive movements), the species adopting both slower diffusion and advection can be favorable or unfavorable depending on the gap between two diffusion rates. When the downstream loss is small (incurred by water flow alone), the species adopting a combination of faster diffusion and weak advection will persist while the movement of faster diffusion and advection can be favorable or unfavorable for species to survive depending on the gap between two advection rates.

Coexistence for a population model with forest fire epidemics

We investigate the effect on survival and coexistence of introducing forest fire epidemics to a certain two-species competition model. The model is an extension of the one introduced by Durrett and Remenik (Ann. Appl. Probab. 19 (2009) 1656–1685), who studied a discrete time particle system running on a random 3-regular graph where occupied sites grow until they become sufficiently dense so that an epidemic wipes out large clusters. In our extension we let two species affected by independent epidemics compete for space, and we allow the epidemic to attack not only giant clusters, but also clusters of smaller order. Our main results show that there are explicit parameter regions where either one species dominates or there is coexistence; this contrasts with the behavior of the model without epidemics, where the fitter species always dominates. We also discuss the survival and extinction regimes for the model with a single species. In both cases we prove convergence to explicit dynamical systems; simulations suggest that their orbits present chaotic behavior.

Quantitative Characteristics of Stabilizing and Equalizing Mechanisms.

AbstractAn understanding of the mechanisms that facilitate coexistence in ecological communities poses a major challenge to theoretical ecology. A popular paradigmatic scheme distinguishes between two qualitatively different processes that help species to coexist: stabilizing mechanisms increase niche differentiation, making the intraspecific competition stronger than the interspecific one, while equalizing mechanisms diminish fitness differences, making the competition less decisive. Here, we provide an analytic and numeric examination of the quantitative features associated with this scheme for a simple, two-species competition model. We show that the main metrics of persistence change only slightly along the stabilizing-equalizing continuum, where niche overlap increases while fitness differences decreases. Therefore, persistence properties cannot indicate the dominant mechanism that promotes coexistence and vice versa. Cross correlations between abundance time series are shown to provide a decent characterization of the mechanisms that promote coexistence. The relevance of these insights to the analysis of diverse assemblages is discussed.

Fixation in the stochastic Lotka-Volterra model with small fitness trade-offs.

We study the probability of fixation in a stochastic two-species competition model. By identifying a naturally occurring fast timescale, we derive an approximation to the associated backward Kolmogorov equation that allows us to obtain an explicit closed form solution for the probability of fixation of either species. We use our result to study fitness tradeoff strategies and show that, despite some tradeoffs having nearly negligible effects on the corresponding deterministic dynamics, they can have large implications for the outcome of the stochastic system.

Asymptotic behavior of solutions of an ODE-PDE hybrid competition system

We study the large-time behavior of a two-species competition model in a spatially heterogeneous environment and investigate the influence of dispersal strategy on the competition. In particular, we allow one species to exhibit a random dispersal movement while the second species is constrained to a non-spatial movement dynamic. We show that there is a nonnegative threshold number (possibly equal to infinity and determined by both species' local intrinsic growth rates) such that the species adopting the random dispersal strategy persists uniformly in space if its diffusion rate is kept below this number while it goes extinct if its diffusion rate is greater than this number. Furthermore, in the critical case that the diffusing species' movement rate equals this number, exactly two scenarios are possible: the non-diffusing species wins the competition if it has a sink area; otherwise, both species coexist.

Two-species competition model with chemotaxis: well-posedness, stability and dynamics

We study a system of PDEs modelling the population dynamics of two competitive species whose spatial movements are governed by both diffusion and mutually repulsive chemotaxis effects. We prove that solutions to this system are globally well-posed, without any smallness assumptions on the chemotactic coefficients. Moreover, in the weak competition regime, we prove that neither species can be driven to extinction as the time goes to infinity, regardless of how strong the chemotaxis coefficients are. Finally, long-time behaviours of the system are studied both analytically in the weakly nonlinear regime, and numerically in the fully nonlinear regime.

Allee effect in models of interacting species

The search for more realistic models for interacting species has produced many adaptations of the original Lotka-Volterra equations, such as the inclusion of the Allee effect and the different Holling’s types of functional response. In the present work we show that a correct implementation of both ideas together requires a careful formulation. We focus our work in the fact that a density dependent carrying capacity, combined with the Allee effect, can lead to meaningless effects. We illustrate the difficulties in predator-prey and two-species competition models, together with our proposed solution of the careful inclusion of the corresponding cubic terms.

Dynamical behaviors of a classical Lotka–Volterra competition–diffusion–advection system

This paper deals with a two-species competition model in a homogeneous advective environment, where two species are subjected to a net loss of individuals at the downstream end. Under the assumption that the advection and diffusion rates of two species are proportional, we give a basic classification on the global dynamics by employing the theory of monotone dynamical system. It turns out that bistability does not happen, but coexistence and competitive exclusion may occur. Furthermore, we present a complete classification on the global dynamics in terms of the growth rates of two species. However, once the above assumption does not hold, bistability may occur. In detail, there exists a tradeoff between growth rates of two species such that competition outcomes can shift between three possible scenarios, including competitive exclusion, bistability and coexistence. These results show that growth competence is important to determine dynamical behaviors.

Stability results for a nonlinear two-species competition model with size-structure

We formulate a system of integro-differential equations to model the dynamics of competition in a two-species community, in which the mortality, fertility and growth are size-dependent. Existence and uniqueness of nonnegative solutions to the system are analyzed. The existence of the stationary size distributions is discussed, and the linear stability is investigated by means of the semigroup theory of operators and the characteristic equation technique. Some sufficient conditions for asymptotical stability / instability of steady states are obtained. The resulting conclusion extends some existing results involving age-independent and age-dependent population models.

Bistability of a two-species competition model with stage structure

In this paper, we focus on the dynamic behaviors of the bistability for a two-species Lotka–Volterra competition model with stage structure. Using the theory of generalized saddle-point behaviors, it is shown that there exists a [Formula: see text]-separatrix [Formula: see text] on its state space such that Species 2 wins the competition whenever the initial distribution is above [Formula: see text], while Species 1 wins the competition whenever the initial distribution is below [Formula: see text]. Combining with the previous conclusions of the system, we give a complete classification for global competitive dynamics. Furthermore, we carefully analyze the changes of the competition outcome after introducing the stage structure. Finally, some numerical simulations are provided to illustrate the effectiveness of the theoretical results.

Hopf bifurcation in a reaction-diffusive-advection two-species competition model with one delay

In this paper, we investigate a reaction-diffusive-advection two-species competition model with one delay and Dirichlet boundary conditions. The existence and multiplicity of spatially non-homogeneous steady-state solutions are obtained. The stability of spatially nonhomogeneous steady-state solutions and the existence of Hopf bifurcation with the changes of the time delay are obtained by analyzing the distribution of eigenvalues of the infinitesimal generator associated with the linearized system. By the normal form theory and the center manifold reduction, the stability and bifurcation direction of Hopf bifurcating periodic orbits are derived. Finally, numerical simulations are given to illustrate the theoretical results.

Analysis of Coexistence and Extinction in a Two-Species Competition Model

We study a competition model of two competing species in population biology having exponential and rational growth functions described by Alexander et al. [1992]. They observed that, for some choice of parameters, the competition model has a chaotic attractor [Formula: see text] for which the basin of attraction is riddled. Here, we give a detailed analysis to illustrate what happens when the normal parameter in this model changes. In fact, by varying the normal parameter, we discuss how the geometry of the basin of attraction of [Formula: see text], the region of coexistence or extinction, changes and illustrate the transitions between the set [Formula: see text] being an asymptotically stable attractor (extinction of rational species), a locally riddled basin attractor and a normally repelling chaotic saddle (extinction of exponential species). Additionally, we show that the riddling and the blowout bifurcation occur. Numerical simulations are presented graphically to confirm the validity of our results. In particular, we verify the occurrence of synchronization for some values of parameters. Finally, we apply the uncertainty exponent and the stability index to quantify the degree of riddling basin. Our observation indicates that the stability index is positive for Lebesgue for almost all points whenever the riddling occurs.

Positive solution branches of two-species competition model in open advective environments

The effect of competition is an important topic in spatial ecology. This paper deals with a general two-species competition system in open advective and inhomogeneous environments. At first, the critical values on the interspecific competition coefficients are established, which determine the stability of semi-trivial steady states. Secondly, by analyzing the nonexistence of coexistence steady states and using the theory of monotone dynamical system, we find that the competitive exclusion principle holds if one of the interspecific competition coefficients is large and the other is in a certain range. Thirdly, in terms of these critical values, the structure and direction of bifurcating branches of positive equilibria arising from two semi-trivial steady states are given by means of the bifurcation theory and stability analysis. Finally, we show that multiple coexistence occurs under certain regimes.