Bifurcation analysis of a predator-prey system with self- and cross-diffusion and constant harvesting rate

In this paper, we focus on a ratio dependent predator-prey system with self- and cross-diffusion and constant harvesting rate. By making use of a brief stability and bifurcation analysis, we derive the symbolic conditions for Hopf, Turing and wave bifurcations of the system in a spatial domain. Additionally, we illustrate spatial pattern formations caused by these bifurcations via numerical examples. A series of numerical examples reveal that one can observe several typical spatiotemporal patterns such as spotted, spot-stripelike mixtures due to Turing bifurcation and an oscillatory wave pattern due to the wave bifurcation. Thus the obtained results disclose that the spatially extended system with self-and cross-diffusion and constant harvesting rate plays an important role in the spatiotemporal pattern formations in the two dimensional space

Existence of spatial patterns in reaction–diffusion systems incorporating a prey refuge

In real-world ecosystem, studies on the mechanisms of spatiotemporal pattern formation in a system of interacting populations deserve special attention for its own importance in contemporary theoretical ecology. The present investigation deals with the spatial dynamical system of a two-dimensional continuous diffusive predator–prey model involving the influence of intra-species competition among predators with the incorporation of a constant proportion of prey refuge. The linear stability analysis has been carried out and the appropriate condition of Turing instability around the unique positive interior equilibrium point of the present model system has been determined. Furthermore, the existence of the various spatial patterns through diffusion-driven instability and the Turing space in the spatial domain have been explored thoroughly. The results of numerical simulations reveal the dynamics of population density variation in the formation of isolated groups, following spotted or stripe-like patterns or coexistence of both the patterns. The results of the present investigation also point out that the prey refuge does have significant influence on the pattern formation of the interacting populations of the model under consideration

Dinámica estocástica en física y biología

Tesis de la Universidad Complutense de Madrid, Facultad de Ciencias Físicas, Departamento de Estructura de la Materia, Física Térmica y Electrónica, leída el 26-10-2018Stochastic processes are present in virtually any field of science. Actually, in general deterministic processes are no more than an approximation of more complex stochastic processes in some regime of validity. In the present thesis, we will study some systems that appear in Physics and Biology where the presence of these stochastic processes plays a major role. First, we will study the Brownian ratchet systems, which are able to generate a directed motion simply by rectifying the thermal fluctuations to which Brownian particles are subjected. These Brownian ratchets began to be studied in the field of Statistical Physics as a Gedanken experiment that apparently broke the Second Law of Thermodynamics. Later they were applied to study the operation of some molecular motors. In this thesis we will study different types of Brownian ratchets, characterizing the average flux, the efficiency, and the quality of the transport of particles they produce..En prácticamente cualquier rama de la ciencia están presentes los procesos estocásticos. De hecho, en general los procesos deterministas no son más que una aproximación de procesos estocásticos más complejos en algún régimen de validez. En la presente tesis, estudiaremos algunos sistemas que aparecen en Física y Biología donde la presencia de dichos procesos es crucial.Primero, estudiaremos los conocidos como trinquetes brownianos, que son capaces de generar movimiento dirigido simplemente a través de la rectificación de las fluctuaciones térmicas a las que se ven sometidas las partículas Brownianas. Estos trinquetes Brownianos se empezaron a estudiar en el campo de la Física Estadística como un experimento mental que aparentemente rompía la Segunda Ley de la Termodinámica. Sin embargo, más adelante se ha visto que pueden ser aplicados para estudiar el funcionamiento de algunos motores moleculares. En esta tesis estudiaremos distintos tipos de trinquetes Brownianos, caracterizando el flujo medio, la eficiencia, y la calidad del transporte de partículas que producen...Depto. de Estructura de la Materia, Física Térmica y ElectrónicaFac. de Ciencias FísicasTRUEunpu

Moving forward in circles: challenges and opportunities in modelling population cycles

Population cycling is a widespread phenomenon, observed across a multitude of taxa in both laboratory and natural conditions. Historically, the theory associated with population cycles was tightly linked to pairwise consumer–resource interactions and studied via deterministic models, but current empirical and theoretical research reveals a much richer basis for ecological cycles. Stochasticity and seasonality can modulate or create cyclic behaviour in non-intuitive ways, the high-dimensionality in ecological systems can profoundly influence cycling, and so can demographic structure and eco-evolutionary dynamics. An inclusive theory for population cycles, ranging from ecosystem-level to demographic modelling, grounded in observational or experimental data, is therefore necessary to better understand observed cyclical patterns. In turn, by gaining better insight into the drivers of population cycles, we can begin to understand the causes of cycle gain and loss, how biodiversity interacts with population cycling, and how to effectively manage wildly fluctuating populations, all of which are growing domains of ecological research

Optimal harvesting policy of a prey–predator model with Crowley–Martin-type functional response and stage structure in the predator

In this paper, a three-dimensional dynamical model consisting of a prey, a mature predator, and an immature predator is proposed and analysed. The interaction between prey and mature predator is assumed to be of the Crowley–Martin type, and both the prey and mature predator are harvested according to catch-per-unit-effort (CPUE) hypothesis. Steady state of the system is obtained, stability analysis (local and global both) are discussed to explore the long-time behaviour of the system. The optimal harvesting policy is also discussed with the help of Pontryagin's maximum principle. The harvesting effort is taken as an effective control instrument to preserve prey and predator and to maintain them at an optimal level

Turing instability in a diffusive predator-prey model with multiple Allee effect and herd behavior

Diffusion-driven instability and bifurcation analysis are studied in a predator-prey model with herd behavior and quadratic mortality by incorporating multiple Allee effect into prey species. The existence and stability of the equilibria of the system are studied. And bifurcation behaviors of the system without diffusion are shown. The sufficient and necessary conditions for Turing instability occurring are obtained. And the stability and the direction of Hopf and steady state bifurcations are explored by using the normal form method. Furthermore, some numerical simulations are presented to support our theoretical analysis. We found that too large diffusion rate of prey prevents Turing instability from emerging. Finally, we summarize our findings in the conclusion

Temporal and spatial patterns in a diffusive ratio-dependent predator–prey system with linear stocking rate of prey species

The ratio-dependent predator–prey model exhibits rich interesting dynamics due to the singularity of the origin. It is one of prototypical pattern formation models. Stocking in a ratio-dependent predator–prey models is relatively an important research subject from both ecological and mathematical points of view. In this paper, we study the temporal, spatial patterns of a ratio-dependent predator–prey diffusive model with linear stocking rate of prey species. For the spatially homogeneous model, we derive conditions for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solution by the center manifold and the normal form theory. For the reaction-diffusion model, firstly it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Then it is demonstrated that the model exhibits Hopf bifurcation which produces temporal inhomogeneous patterns. Finally, the non-existence and existence of positive non-constant steady-state solutions are established. We can see spatial inhomogeneous patterns via Turing instability, temporal periodic patterns via Hopf bifurcation and spatial patterns via the existence of positive non-constant steady state. Moreover, numerical simulations are performed to visualize the complex dynamic behavior

Temporal and spatial patterns in a diffusive ratio-dependent predator-prey system with linear stocking rate of prey species

The ratio-dependent predator–prey model exhibits rich interesting dynamics due to the singularity of the origin. It is one of prototypical pattern formation models. Stocking in ratio-dependent predator–prey models is relatively an important research subject from both ecological and mathematical points of view. In this paper, we study the temporal, spatial patterns of a ratio-dependent predator–prey diffusive model with linear stocking rate of prey species. For the spatially homogeneous model, we derive conditions for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solution by the center manifold and the normal form theory. For the reaction-diffusion model, firstly it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Then it is demonstrated that the model exhibits Hopf bifurcation which produces temporal inhomogeneous patterns. Finally, the non-existence and existence of positive non-constant steady-state solutions are established. We can see spatial inhomogeneous patterns via Turing instability, temporal periodic patterns via Hopf bifurcation and spatial patterns via the existence of positive non-constant steady state. Moreover, numerical simulations are performed to visualize the complex dynamic behavior

Balanced exploitation and coexistence of interacting, size-structured, fish species

This paper examines some effects of exploitation on a simple ecosystem containing two interacting fish species, with life histories similar to mackerel (Scomber scombrus) and cod (Gadus morhua), using a dynamic, size-spectrum model. Such models internalize body growth and mortality from predation, allowing bookkeeping of biomass at a detailed level of individual predation and growth and enabling scaling up to the mass balance of the ecosystem. Exploitation set independently for each species with knife-edge, size-at-entry fishing can lead to collapse of cod. Exploitation to achieve a fixed ratio of yield to productivity across species can also lead to collapse of cod. However, harvesting balanced to the overall productivity of species in the exploited ecosystem exerts a strong force countering such collapse. If balancing across species is applied to a fishery with knife-edge selection, size distributions are truncated, changing the structure of the system and reducing its resilience to perturbations. If balancing is applied on the basis of productivity at each body size as well as across species, there is less disruption to size-structure, resilience is increased, and substantially greater biomass yields are possible. We note an identity between the body size at which productivity is maximized and the age at which cohort biomass is maximized. In our numerical results based on detailed bookkeeping of biomass, cohort biomass reaches its maximum at body masse