Choosing space Cg as the phase space, the existence, uniqueness and stability of the solution to neutral stochastic functional differential equations with infinite delay (short for INSFDEs) are studied in this paper. Under non-Lipschitz condition, weakened linear growth condition and contractive condition, the existence-and-uniqueness theorem of the solution to INSFDEs by means of the Picard iteration, Doob’s martingale inequalities, Gronwall’s inequality and Bihari’s inequality is obtained. Furthermore, the continuous dependence of the solutions on the initial value to INSFDEs are derived.

We study resident-invader dynamics in fluctuating environments when the invader and the resident have close but distinct strategies. First we focus on a class of continuous-time models of unstructured populations of multi-dimensional strategies, which incorporates environmental feedback and environmental stochasticity. Then we generalize our results to a class of structured population models. We classify the generic population dynamical outcomes of an invasion event when the resident population in a given environment is non-growing on the long-run and stochastically persistent. Our approach is based on the series expansion of a model with respect to the small strategy difference, and on the analysis of a stochastic fast-slow system induced by time-scale separation. Theoretical and numerical analyses show that the total size of the resident and invader population varies stochastically and dramatically in time, while the relative size of the invader population changes slowly and asymptotically in time. Thereby the classification is based on the asymptotic behavior of the relative population size, and which is shown to be fully determined by invasion criteria (i.e., without having to study the full generic dynamical system). Our results extend and generalize previous results for a stable resident equilibrium (particularly, Geritz in J Math Biol 50(1):67–82, 2005; Dercole and Geritz in J Theor Biol 394:231-254, 2016) to non-equilibrium resident population dynamics as well as resident dynamics with stochastic (or deterministic) drivers.

We study the adaptive dynamics of the colonization rate of species living in a patchy habitat when there is a trade-off with the competitive strength for individual patches. To that end, we formulate a continuous-time competition-colonization model that also includes ownership effects as well as random disturbance affecting the mortality rate. We find that intermediate disturbance (as measured by the fluctuation intensity of the mortality rate), a strong competition-colonization trade-off, and a weak ownership effect are necessary conditions for evolutionary branching and hence for the emergence of polymorphisms (i.e., coexistence) by small evolutionary steps. Specifically, concerning ownership we find that with low-intermediate disturbance, a weak ownership advantage favours evolutionary branching while ownership disadvantage does not. This asymmetry disappears at the higher-intermediate disturbance. Moreover, at a low-intermediate disturbance, the effect of the strength of the competition-colonization trade-off on evolutionary branching is non-monotonic disappears because the possibility of branching disappears again when the trade-off is too strong. We also find that there can be multiple evolutionary attractors for polymorphic populations, each with its own basin of attraction. With small but non-zero random evolutionary steps and depending on the initial polymorphic condition just after branching, a coevolutionary trajectory may come arbitrarily close to the shared boundary of two such basins and may even jump from one side to the other, which can lead to various kinds of long-term evolutionary dynamics, including evolutionary branching-extinction cycles.

A kind of general stochastic nonautonomous Lotka-Volterra models with infinite delay is investigated in this paper. By constructing several suitable Lyapunov functions, the existence and uniqueness of global positive solution and global asymptotic stability are obtained. Further, the solution asymptotically follows a normal distribution by means of linearizing stochastic differential equation. Moment estimations in time average are derived to improve the approximation distribution. Finally, numerical simulations are given to illustrate our conclusions.