How Landscape Heterogeneity Frames Optimal Diffusivity in Searching Processes

Theoretical and empirical investigations of search strategies typically have failed to distinguish the distinct roles played by density versus patchiness of resources. It is well known that motility and diffusivity of organisms often increase in environments with low density of resources, but thus far there has been little progress in understanding the specific role of landscape heterogeneity and disorder on random, non-oriented motility. Here we address the general question of how the landscape heterogeneity affects the efficiency of encounter interactions under global constant density of scarce resources. We unveil the key mechanism coupling the landscape structure with optimal search diffusivity. In particular, our main result leads to an empirically testable prediction: enhanced diffusivity (including superdiffusive searches), with shift in the diffusion exponent, favors the success of target encounters in heterogeneous landscapes

Nonlocal reaction–diffusion models of heterogeneous wealth distribution

Dynamics of human populations can be affected by various socio-economic factors through their influence on the natality and mortality rates, and on the migration intensity and directions. In this work we study an economic–demographic model which takes into account the dependence of the wealth production rate on the available resources. In the case of nonlocal consumption of resources, the homogeneous-in-space wealth–population distribution is replaced by a periodic-inspace distribution for which the total wealth increases. For the global consumption of resources, if the wealth redistribution is small enough, then the homogeneous distribution is replaced by a heterogeneous one with a single wealth accumulation center. Thus, economic and demographic characteristics of nonlocal and global economies can be quite different in comparison with the local economy

Long transients in discontinuous time-discrete models of population dynamics

Traditionally, mathematical modelling of population dynamics was focused on long-term, asymptotic behaviour (systems attractors), whereas the effects of transient regimes were largely disregarded. However, recently there has been a growing appreciation of the role of transients both in empirical ecology and theoretical studies. Among the main challenges are identification of the mechanisms triggering transients in various dynamical systems and understanding of the corresponding scaling law of the transient's lifetime; the latter is of a vital practical importance for long-term ecological forecasting and regime shifts anticipation. In this study, we reveal and investigate various patterns of long transients occurring in two generic time-discrete population models which are mathematically described by discontinuous (piece-wise) maps. In particular, we consider a single-species population model and a predator–prey system, in each model we assume that the dispersal of species at the end of each season is density dependent. For both models, we demonstrate transients due to crawl-by dynamics, chaotic repellers, chaotic saddles, ghost attractors, and a rich variety of intermittent regimes. For each type of transient, we investigate the corresponding scaling law of the transient's lifetime. We explore the space of key model parameters, to find where particular types of long transients can be expected, and we show that long transients are omnipresent since they can be observed within a wide range of model parameters. We also reveal the possibility of complex patterns occurring as a cascade of transients of different types. We also considered a stochastic version of the model where some parameters exhibit random fluctuations. We show that stochasticity can reduce, extend or produce new patterns of long transients. We conclude that the discontinuity in population models significantly facilitates the emergence of long transients by creating new types and increasing parameter domains of the corresponding transient dynamics. Another important conclusion is that the asymptotic regime of population dynamics is hardly possible to predict based on a finite time course of species densities, which is crucial for ecosystem management and decision making

Regime shifts, extinctions and long transients in models of population dynamics with density-dependent dispersal

Predicting extinctions resulting from ecosystems' regime shifts has long been a focus of biological conservation and ecological management. Mathematical modelling plays a key role in assessing the possibility of such events. Traditionally, however, models focused on long-term, asymptotic behaviour of ecosystems. Meanwhile, the environment is usually non-stationary, which may mean that the long-term behaviour is never observed. Correspondingly, over the last two decades there has been a growing appreciation of the role of transients both in empirical ecology and theoretical studies, in particular in the context of species extinctions. In this paper, we theoretically explore long transients and extinctions occurring in several paradigmatic models of increasing complexity, such as single species, two-species and three-species systems. We consider the population dynamics in a local ‘patch’, the patch being connected to the rest of the population via density-dependent dispersal. We consider both deterministic and stochastic scenarios. We discover many different patterns of long transient dynamics with quick regime shifts between ‘safe’ (persistent) dynamics and unsafe ones resulting in extinctions. Remarkably, delayed extinction can occur after thousands of generations of apparently safe population dynamics. We classify transient regimes and reveal their underlying mechanisms. Environmental noise can either shorten transients or can create a new type of a long transient. Our study suggests that not only are long transients ubiquitous, but there is also a great variety of them. The omnipresence of long transients emphasizes the need to account for them in nature conservation programs as well as future theoretical research.</p