Permanence and Asymptotically Stable Complete Trajectories for Nonautonomous Lotka–Volterra Models with Diffusion

Abstract

Abstract. Lotka-Volterra systems are the canonical ecological models used to analyze population dynamics of competition, symbiosis or prey-predator behaviour involving different interacting species in a fixed habitat. Much of the work on these models has been within the framework of infinite-dimensional dynamical systems, but this has frequently been extended to allow explicit time dependence, generally in a periodic, quasiperiodic or almost periodic fashion. The presence of more general non-autonomous terms in the equations leads to non-trivial difficulties which have stalled the development of the theory in this direction. However, the theory of non-autonomous dynamical systems has received much attention in the last decade, and this has opened new possibilities in the analysis of classical models with general non-autonomous terms. In this paper we use the recent theory of attractors for non-autonomous PDEs to obtain new results on the permanence and the existence of forwards and pullback asymptotically stable global solutions associated to non-autonomous Lotka-Volterra systems describing competition, symbiosis or prey-predator phenomena. We note in particular that our results are valid for prey-predator models, which are not order-preserving: even in the 'simple' autonomous case the uniqueness and global attractivity of the positive equilibrium (which follows from the more general results here) is new.

Uniform and strict persistence in monotone skew-product semiflows with applications to non-autonomous Nicholson systems

“…The papers by Faria and Röst [12], Freedman and Ruan [15], Garay and Hofbauer [16], Hetzer and Shen [20], Hirsch et al [21], Langa et al [26], Magal and Zhao [28], Mierczyński and Shen [30], Mierczyński et al [31], Novo et al [36], Salceanu and Smith [43], Schreiber [44,45], Thieme [51,52], Wang and Zhao [54], and references therein, provide a long but not complete list of works on this topic.…”

Structure and bifurcation of pullback attractors in a non-autonomous Chafee-Infante equation

“…As a direct application of the results in [11] we obtain the following description of the pullback attractor within the positive cone. As a consequence, this result also provides an example of a 'non-autonomous pitchfork bifurcation' in the sense of [10].…”

“…The existence of global pullback attractors for this equation is known (see, for instance, [14]). Moreover, the results in [11,13] provide the following upper and lower bounds for bounded global solutions to (3.1). In particular, the pullback attractor lies between two 'extremal bounded solutions'.…”

Persistence in Forward Nonautonomous Competitive Systems of Parabolic Equations

“…In [22], the authors of the current paper studied the persistence for general nonautonomous and random cases by applying the principal spectral theory developed in [22]. The authors of [15] provided various sufficient conditions for uniform persistence in quite general nonautonomous cases. In [24], uniform persistence for nonautonomous n-species competition system is studied.…”

“…Uniform persistence for autonomous and time periodic nonlinear systems of parabolic partial differential equations of second order has been well studied (see [1][2][3][4][5][6]12,16,28], etc.). It has also been recently intensively studied for various nonautonomous nonlinear systems of parabolic equations (see [8], [15,22,24,26,27], etc.). We point out that in [8], Hetzer and Shen gave a study for time almost periodic cases.…”