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An ecological system with multiple stable equilibria is prone to undergo catastrophic change or regime shift from one steady-state to another. It should be noted that, if one of the steady states is an extinction state, the catastrophic…
We study the spatial patterns formed by a system of interacting particles where the mobility of any individual is determined by the population crowding at two different spatial scales. In this way we model the behavior of some biological…
The study of density-dependent stochastic population processes is important from a historical perspective as well as from the perspective of a number of existing and emerging applications today. In more recent applications of these…
We study a stochastic linear discrete metapolulation model to understand the effect of risk spreading by dispersion. We calculate analytically the stable distribution of populations that live in different habitats. The result shows that the…
In this paper exponential stability of nonlinear fractional order stochastic system with Poisson jumps is studied in finite dimensional space. Existence and uniqueness of solution, stability and exponential stability results are established…
The existence of global nonnegative martingale solutions to cross-diffusion systems of Shigesada-Kawasaki-Teramoto type with multiplicative noise is proven. The model describes the stochastic segregation dynamics of an arbitrary number of…
For cross-diffusion systems possessing an entropy (i.e. a Lyapunov functional)we study nonlocal versions and exhibit sufficient conditions to ensure that thenonlocal version inherits the entropy structure. These nonlocal systems can…
Models of population growth and extinction are an increasingly popular subject of study. However, consequences of stochasticity and noise in shaping distributions and outcomes are not sufficiently explored. Here we consider a distributed…
An ab-initio numerical study of the density-dependent, evolutionary stable dispersal strategy is presented. The simulations are based on a simple discretei generation island model with four processes: reproduction, dispersal, competition…
Since the early 1970s, stellar population modelling has been one of the basic tools for understanding the physics of unresolved systems from observation of their integrated light. Models allow us to relate the integrated spectra (or…
I discuss the so-called stochastic individual based model of adaptive dynamics and in particular how different scaling limits can be obtained by taking limits of large populations, small mutation rate, and small effect of single mutations…
Populations interact non-linearly and are influenced by environmental fluctuations. In order to have realistic mathematical models, one needs to take into account that the environmental fluctuations are inherently stochastic. Often,…
We study the evolution of interacting groups of agents in two-dimensional geometries. We introduce a microscopic stochastic model that includes floor fields modeling the global flow of individual groups as well as local interaction rules.…
Progress on modern scientific questions regularly depends on using large-scale datasets to understand complex dynamical systems. An especially challenging case that has grown to prominence with advances in single-cell sequencing…
This work deals with two problems arising in mathematical ecology. The first problem is concerned with diploid branching particle models and its behavior when rapid stirring is added to the interaction. The particle models involve two types…
In this paper a stochastic reaction diffusion system is considered, which models the spread of a finite population reacting with a non-renewable resource in the presence of individual based noise. A two-parameter phase diagram is…
Diffusion models accomplish remarkable success in data generation tasks across various domains. However, the iterative sampling process is computationally expensive. Consistency models are proposed to learn consistency functions to map from…
Nonlocal aggregation-diffusion models, when coupled with a spatial map, can capture cognitive and memory-based influences on animal movement and population-level patterns. In this work, we study a one-dimensional…
Many approaches to modelling reaction-diffusion systems with anomalous transport rely on deterministic equations and ignore fluctuations arising due to finite particle numbers. Starting from an individual-based model we use a…
The question of biological stability (permanence) of a replicator reaction-diffusion system is considered. Sufficient conditions of biological stability are found. It is proved that there are situations when biologically unstable…