Analysis of a Single Species Model with Dissymmetric Bidirectional Impulsive Diffusion and Dispersal Delay

In the natural ecosystem, impulsive diffusion provides a more natural description for population dynamics. In addition, dispersal processes often involve with time delay. In view of these facts, a single species model with impulsive diffusion and dispersal delay is formulated. By the stroboscopic map of the discrete dynamical system and other analysis methods, the permanence of the system is investigated. Moreover, sufficient conditions on the existence and uniqueness of a positive periodic solution for the system are derived from the intermediate value theorem. We also demonstrate the global stability of the positive periodic solution by the theory of discrete dynamical system. Finally, numerical simulations and discussion are presented to validate our theoretical results.

In most models of population dynamics, diffusion between two patches is assumed to be either continuous or discrete. However, in the real world, it is often the case that diffusion occurs at certain moment every year, impulsive diffusion can provide a more suitable manner to model the actual dispersal (or migration) behaviors for many ecological species. In addition, it is generally recognized that some kinds of time delays are inevitable in population interactions. In view of these facts, a delayed predator–prey system with impulsive diffusion between two patches is proposed. By using comparison theorem of impulsive differential equation and some analysis techniques, criteria on the global attractivity of predator-extinction periodic solution are established, sufficient conditions for the permanence of system are obtained. Finally, numerical simulations are presented to illustrate our theoretical results.

In most models of population dynamics, diffusion between patches is assumed to be continuous or discrete, but in practice many species diffuse only during a single period. In this paper we propose a single species model with impulsive diffusion between two patches, which provides a more natural description of population dynamics. By using the discrete dynamical system generated by a monotone, concave map for the population, we prove that the map always has a globally stable positive fixed point. This means that a single species system with impulsive diffusion always has a globally stable positive periodic solution. This result is further substantiated by numerical simulation. Under impulsive diffusion the single species survives in the two patches.

Single species models with logistic growth and dissymmetric impulse dispersal

Species distribution models (SDM) are a key tool in ecology, conservation and management of natural resources. Two key components of the state-of-the-art SDMs are the description for species distribution response along environmental covariates and the spatial random effect. Joint species distribution models (JSDMs) additionally include interspecific correlations which have been shown to improve their descriptive and predictive performance compared to single species models. Current JSDMs are restricted to hierarchical generalized linear modeling framework. These parametric models have trouble in explaining changes in abundance due, e.g., highly non-linear physical tolerance limits which is particularly important when predicting species distribution in new areas or under scenarios of environmental change. On the other hand, semi-parametric response functions have been shown to improve the predictive performance of SDMs in these tasks in single species models. Here, we propose JSDMs where the responses to environmental covariates are modeled with additive multivariate Gaussian processes coded as linear models of coregionalization. These allow inference for wide range of functional forms and interspecific correlations between the responses. We propose also an efficient approach for inference with Laplace approximation and parameterization of the interspecific covariance matrices on the euclidean space. We demonstrate the benefits of our model with two small scale examples and one real world case study. We use cross-validation to compare the proposed model to analogous semi-parametric single species models and parametric single and joint species models in interpolation and extrapolation tasks. The proposed model outperforms the alternative models in all cases. We also show that the proposed model can be seen as an extension of the current state-of-the-art JSDMs to semi-parametric models.

In this paper, we consider a predator-prey model with prey impulsive diffusion and dispersal delay. By utilizing the dynamical properties of a single-species model with diffusion and dispersal delay between two patches and the comparison principle of impulsive differential equations, we establish the sufficient conditions on the global attractivity of predator-extinction periodic solution and the permanence of species for the model.

In this paper, a single species model with impulsive diffusion between two patches is proposed, which provides a more natural description of plant seeds dynamics compared with the continuous or discrete ones. By using the discrete dynamical system generated by a monotone, concave map for the dispersal model and a ∊1 - ∊2 variation, it is proved that the Poincare map has a globally stable positive fixed point. This implies that the system considered here has a globally stable positive periodic solution under some sufficient conditions. Further numerical simulations show that the diffusion can save the extinction though the species has a negative growth rate in one patch.

A single species model with impulsive diffusion and pulsed harvesting

The reverse effects of random perturbation on discrete systems for single and multiple population models

We establish a class of intermittentbidirectional dispersal population models with almost periodicparameters and dispersal delays between two patches. The form ofdispersal discussed in this paper is different from bothcontinuous and impulsive dispersals, in which the dispersalbehavior occurs either in a sustained manner or instantaneously;instead, it is a synthesis of these types. Dynamical propertiessuch as permanence, existence, uniqueness, and globally asymptoticstability of almost periodic solutions are investigated by usingLiapunov-Razumikhin type technique, using the comparison theorem,constructing a suitable Lyapunov functional, using almost periodicfunctional hull theory and analysis approach, etc. Finally,numerical simulations are presented and discussed to illustrate our analytic results,by which we find that intermittent dispersal systems are more complicated than continuous or impulsive dispersal systems.

Mathematical analysis of a delayed stage-structured predator–prey model with impulsive diffusion between two predators territories

Impulsive diffusion in single species model

<p style='text-indent:20px;'>This paper investigates the effect of environmental heterogeneity on species spreading via numerical simulation of suitable reaction-diffusion models with free boundaries. We focus on the changes of long-time dynamics (establishment or extinction) and spreading speeds of the species as the parameters describing the heterogeneity of the environment are varied. For the single species model in time-periodic environment and in space-periodic environment theoretically treated in [<xref ref-type="bibr" rid="b15">15</xref>,<xref ref-type="bibr" rid="b16">16</xref>], we obtain more detailed properties here. Among other results, our numerical simulation suggests that, in a time-periodic or space-periodic environment, moderate increase of the oscillation scale enhances the chances of establishment as well as the spreading speed of the species. We also numerically examine a related model with two competing species, which was treated in [<xref ref-type="bibr" rid="b34">34</xref>,<xref ref-type="bibr" rid="b28">28</xref>,<xref ref-type="bibr" rid="b24">24</xref>] recently and reduces to the single species free boundary model when one of the species is absent. Our numerical results, obtained by varying the parameters in the time-periodic and space-periodic terms of the model, suggest that heterogeneity of the environment enhances the invasion of the two species (as in the single species model), although there are subtle differences of the influences felt by the two. Some intriguing phenomena revealed in our simulations suggest that heterogeneity of the environment decreases the level of predictability of the competition outcome.</p>

Survival analysis of a single-species population model with fluctuations and migrations between patches

Near infra-red spectroscopy quantitative modelling of bivalve protein, lipid and glycogen composition using single-species versus multi-species calibration and validation sets