Spatial proliferation of invasive species often causes serious damage to agriculture, ecology and environment. Evaluation of the extent of the area potentially invadable by an alien species is an important problem. Landscape features that reduces dispersal space to narrow corridors can make some areas inaccessible to the invading species. On the other hand, the existence of stepping stones-small areas or 'patches' with better environmental conditions-is known to assist species spread. How an interplay between these factors can affect the invasion success remains unclear. In this paper, we address this question theoretically using a mechanistic model of population dynamics. Such models have been generally successful in predicting the rate and pattern of invasive spread; however, they usually consider the spread in an unbounded, uniform space hence ignoring the complex geometry of a real landscape. In contrast, here we consider a reaction-diffusion model in a domain of a complex shape combining corridors and stepping stones. We show that the invasion success depends on a subtle interplay between the stepping stone size, location and the strength of the Allee effect inside. In particular, for a stepping stone of a small size, there is only a narrow range of locations where it can unblock the otherwise impassable corridor.

A telegraph equation is believed to be an appropriate model of population dynamics as it accounts for the directional persistence of individual animal movement. Being motivated by the problem of habitat fragmentation, which is known to be a major threat to biodiversity that causes species extinction worldwide, we consider the reaction-telegraph equation (i.e., telegraph equation combined with the population growth) on a bounded domain with the goal to establish the conditions of species survival. We first show analytically that, in the case of linear growth, the expression for the domain's critical size coincides with the critical size of the corresponding reaction-diffusion model. We then consider two biologically relevant cases of nonlinear growth, i.e., the logistic growth and the growth with a strong Allee effect. Using extensive numerical simulations, we show that in both cases the critical domain size of the reaction-telegraph equation is larger than the critical domain size of the reaction-diffusion equation. Finally, we discuss possible modifications of the model in order to enhance the positivity of its solutions.

Abstract. This study aims to explore the ways in which population dynamics are affected by the shape and size of fragmented habitats. Habitat fragmentation has become a key concern in ecology over the past 20 years as it is thought to increase the threat of extinction for a number of plant and animal species; particularly those close to the fragment edge. In this study, we consider this issue using mathematical modelling and computer simulations in several domains of various shape and with different strength of the Allee effect. A two-dimensional reaction-diffusion equation (taking the Allee effect into account) is used as a model. Extensive simulations are performed in order to determine how the boundaries impact the population persistence. Our results indicate the following: (i) for domains of simple shape (e.g. rectangle), the effect of the critical patch size (amplified by the Allee effect) is similar to what is observed in 1D space, in particular, the likelihood of population survival is determined by the interplay between the domain size and thee strength of the Allee effect; (ii) in domains of complicated shape, for the population to survive, the domain area needs to be larger than the area of the corresponding rectangle. Hence, it can be concluded that domain size and shape both have crucial effect on population survival.

Patterns and rates of invasive species spread have been a focus of attention for several decades. Majority of studies focused on the species proliferation in a relatively uniform "open space" thus leaving aside the effects of the landscape geometry as given by size and shape of inaccessible areas. In this paper, we address this issue by considering the spatiotemporal dynamics of an alien species in a domain where two large uniform habitats are connected by a narrow corridor. We consider the case where the species is originally introduced into one of the habitats but not to the other. The alien species is assumed to be affected by a predator, so that mathematically our system consists of two coupled diffusion-reaction equations. We show that the corridor tends to slow down the spread: it takes the alien population an extra time to penetrate through the corridor, and this delay time can be significant in the case of patchy spread. We also show that a sufficiently narrow corridor blocks the spread; simple analytical estimates for the critical width of the corridor are obtained. Finally, we show that the corridor can become a refuge for the alien population. If considered on a longer timescale that includes species adaptation and/or climate change, the corridor may then become a source of a secondary invasion.

Very recently, the system of differential equations governing the three-dimensional falling body problem (TDFBP) has been approximately solved. The previously obtained approximate solution was based on the fact that the Earth’s rotation (ER) is quite slow and hence all high order terms of ω in addition to the magnitude ω2R were neglected, where ω is the angular velocity and R is the radius of Earth. However, it is shown in this paper that the ignorance of such magnitudes leads, in many cases, to significant errors in the estimated falling time and other physical quantities. The current results are based on obtaining the exact solutions of the full TDFBP-system and performing several comparisons with the approximate ones in the relevant literature. The obtained results are of great interest and importance, especially for other planets in the Solar System or exterior planets, in which ω and/or ω2R are of considerable amounts and hence cannot be ignored. Therefore, the present analysis is valid in analyzing the TDFBP near to the surface of any spherical celestial body.