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The global dynamics of the two-species Lotka-Volterra competition patch model with asymmetric dispersal is classified under the assumptions of weak competition and the weighted digraph of the connection matrix is strongly connected and cycle-balanced. It is shown that in the long time, either the competition exclusion holds that one species becomes extinct, or the two species reach a coexistence equilibrium, and the outcome of the competition is determined by the strength of the inter-specific competition and the dispersal rates. Our main techniques in the proofs follow the theory of monotone dynamical system and a graph-theoretic approach based on the Tree-Cycle identity.
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Let the population of e.g. a country where some opinion struggle occurs be varying in time, according to Verhulst equation. Consider next some competition between opinions such as the dynamics be described by Lotka and Volterra equations. Two kinds of influences can be used, in such a model, for describing the dynamics of an agent opinion conversion: this can occur (i) either by means of mass communication tools, under some external field influence, or (ii) by means of direct interactions between agents. It results, among other features, that change(s) in environmental conditions can prevent the extinction of populations of followers of some ideology due to different kinds of resurrection effects. The tension arising in the country population is proposed to be measured by an appropriately defined scale index.
We study the dynamics of predator-prey systems where prey are confined to a single region of space and where predators move randomly according to a power-law (Levy) dispersal kernel. Site fidelity, an important feature of animal behaviour, is incorporated in the model through a stochastic resetting dynamics of the predators to the prey patch. We solve in the long time limit the rate equations of Lotka-Volterra type that describe the evolution of the two species densities. Fixing the demographic parameters and the Levy exponent, the total population of predators can be maximized for a certain value of the resetting rate. This optimal value achieves a compromise between over-exploitation and under-utilization of the habitat. Similarly, at fixed resetting rate, there exists a Levy exponent which is optimal regarding predator abundance. These findings are supported by 2D stochastic simulations and show that the combined effects of diffusion and resetting can broadly extend the region of species coexistence in ecosystems facing resources scarcity.
This work is concerned with the existence of entire solutions of the diffusive Lotka-Volterra competition system begin{equation}label{eq:abstract} begin{cases} u_{t}= u_{xx} + u(1-u-av), & qquad xinmathbb{R} cr v_{t}= d v_{xx}+ rv(1-v-bu), & qquad xinmathbb{R} end{cases} quad (1) end{equation} where $d,r,a$, and $b$ are positive constants with $a eq 1$ and $b eq 1$. We prove the existence of some entire solutions $(u(t,x),v(t,x))$ of $(1)$ corresponding to $(Phi_{c}(xi),0)$ at $t=-infty$ (where $xi=x-ct$ and $Phi_c$ is a traveling wave solution of the scalar Fisher-KPP defined by the first equation of $(1)$ when $a=0$). Moreover, we also describe the asymptotic behavior of these entire solutions as $tto+infty$. We prove existence of new entire solutions for both the weak and strong competition case. In the weak competition case, we prove the existence of a class of entire solutions that forms a 4-dimensional manifold.
We use dynamical generating functionals to study the stability and size of communities evolving in Lotka-Volterra systems with random interaction coefficients. The size of the eco-system is not set from the beginning. Instead, we start from a set of possible species, which may undergo extinction. How many species survive depends on the properties of the interaction matrix; the size of the resulting food web at stationarity is a property of the system itself in our model, and not a control parameter as in most studies based on random matrix theory. We find that prey-predator relations enhance stability, and that variability of species interactions promotes instability. Complexity of inter-species couplings leads to reduced sizes of ecological communities. Dynamically evolved community size and stability are hence positively correlated.
The time evolution of a class of completely integrable discrete Lotka-Volterra s ystem is shown not unique but have two different ways chosen randomly at every s tep of generation. This uncertainty is consistent with the existence of constant s of motion and disappears in both continuous time and ultra discrete limits.
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