Ecological system with fear induced group defence and prey refuge

In this study, we investigate the dynamics of a spatial and non spatial prey-predator interaction model that includes the following: (i) fear effect incorporated in prey birth rate; (ii) group defence of prey against predators; and (iii) prey refuge. We provide comprehensive mathematical analysis of extinction and persistence scenarios for both prey and predator species. To better explore the dynamics of the system, a thorough investigation of bifurcation analysis has been performed using fear level, prey birth rate, and prey death rate caused by intra-prey competition as bifurcation parameter. All potential occurrences of bi-stability dynamics have also been investigated for some relevant sets of parametric values. Our numerical evaluations show that high levels of fear can stabilize the prey-predator system by ruling out the possibility of periodic solutions. Also, our model Hopf bifurcation is subcritical in contrast to traditional prey-predator models, which ignore the cost of fear and have supercritical Hopf bifurcations in general. In contrast to the general trend, predator species go extinct at higher values of prey birth rates. We have also found that, contrary to the typical tendency for prey species to go extinct, both prey and predator populations may coexist in the system as intra-prey competition level grows noticeably. The stability and Turing instability of associated spatial model have also been investigated analytically. We also perform the numerical simulation to observe the effect of different parameters on the density distribution of species. Different types of spatiotemporal patterns like spot, mixture of spots and stripes have been observed via variation of time evolution, diffusion coefficient of predator population, level of fear factor and prey refuge. The fear level parameter (k) has a great impact on the spatial dynamics of model system

Dynamical Analysis of a Stochastic Predator-Prey Model with an Allee Effect

We present and analyze a modified Holling type-II predator-prey model that includes some important factors such as Allee effect, density-dependence, and environmental noise. By constructing suitable Lyapunov functions and applying Itô formula, some qualitative properties are given, such as the existence of global positive solutions, stochastic boundedness, and the global asymptotic stability. A series of numerical simulations to illustrate these mathematical findings are presented

Dynamical Analysis of a Stochastic Predator-Prey Model with an Allee Effect

We present and analyze a modified Holling type-II predator-prey model that includes some important factors such as Allee effect, density-dependence, and environmental noise. By constructing suitable Lyapunov functions and applying Itô formula, some qualitative properties are given, such as the existence of global positive solutions, stochastic boundedness, and the global asymptotic stability. A series of numerical simulations to illustrate these mathematical findings are presented

Dynamics of marine zooplankton : social behavior ecological interactions, and physically-induced variability

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution February 2008Marine ecosystems reflect the physical structure of their environment and the biological processes they carry out. This leads to spatial heterogeneity and temporal variability, some of which is imposed externally and some of which emerges from the ecological mechanisms themselves. The main focus of this thesis is on the formation of spatial patterns in the distribution of zooplankton arising from social interactions between individuals. In the Southern Ocean, krill often assemble in swarms and schools, the dynamics of which have important ecological consequences. Mathematical and numerical models are employed to study the interplay of biological and physical processes that contribute to the observed patchiness. The evolution of social behavior is simulated in a theoretical framework that includes zooplankton population dynamics, swimming behavior, and some aspects of the variability inherent to fluid environments. First, I formulate a model of resource utilization by a stage-structured predator population with density-dependent reproduction. Second, I incorporate the predator-prey dynamics into a spatially-explicit model, in which aggregations develop spontaneously as a result of linear instability of the uniform distribution. In this idealized ecosystem, benefits related to the local abundance of mates are offset by the cost of having to share resources with other group members. Third, I derive a weakly nonlinear approximation for the steady-state distributions of predator and prey biomass that captures the spatial patterns driven by social tendencies. Fourth, I simulate the schooling behavior of zooplankton in a variable environment; when turbulent flows generate patchiness in the resource field, schools can forage more efficiently than individuals. Taken together, these chapters demonstrate that aggregation/ schooling can indeed be the favored behavior when (i) reproduction (or other survival measures) increases with density in part of the range and (ii) mixing of prey into patches is rapid enough to offset the depletion. In the final two chapters, I consider sources of temporal variability in marine ecosystems. External perturbations amplified by nonlinear ecological interactions induce transient excursions away from equilibrium; in predator-prey dynamics the amplitude and duration of these transients are controlled by biological processes such as growth and mortality. In the Southern Ocean, large-scale winds associated with ENSO and the Southern Annular Mode cause convective mixing, which in turn drives air-sea fluxes of carbon dioxide and oxygen. Whether driven by stochastic fluctuations or by climatic phenomena, variability of the biogeochemical/physical environment has implications for ecosystem dynamics.Funding was provided by the Academic Programs Office of the MIT-WHOI Joint Program, an Ocean Ventures Fund Award, an Anonymous Ys Endowed Science Fellowship, and by NSF grants OCE-0221369 and OCE-336839

Analysis and simulation on dynamics of a partial differential system with nonlinear functional responses

We introduce a reaction–diffusion system with modified nonlinear functional responses. We first discuss the large-time behavior of positive solutions for the system. And then, for the corresponding steady-state system, we are concerned with the priori estimate, the existence of the nonconstant positive solutions as well as the bifurcations emitting from the positive constant equilibrium solution. Finally, we present some numerical examples to test the theoretical and computational analysis results. Meanwhile, we depict the trajectory graphs and spatiotemporal patterns to simulate the dynamics for the system. The numerical computations and simulated graphs imply that the available food resource for consumer is very likely not single

Modeling the fear effect in the predator-prey dynamics with an age structure in the predators

We incorporate the fear effect and the maturation period of predators into a diffusive predator-prey model. Local and global asymptotic stability for constant steady states as well as uniform persistence of the solution are obtained. Under some conditions, we also exclude the existence of spatially nonhomogeneous steady states and the steady state bifurcation bifurcating from the positive constant steady state. Hopf bifurcation analysis is carried out by using the maturation period of predators as a bifurcation parameter, and we show that global Hopf branches are bounded. Finally, we conduct numerical simulations to explore interesting spatial-temporal patterns

Population dynamics in meerkats, Suricata suricatta

Research on cooperatively breeding species has shown that their population dynamics can differ from those of conventional breeders. Populations of obligate cooperators are structured into social groups, the link between individual behaviour and population dynamics is mediated by group-level demography, and population dynamics can be strongly affected both by features of sociality per se and by resultant population structure. Notably, groups may be subject to inverse density dependence (Allee effects) that result from a dependence on conspecific helpers, but evidence for population-wide Allee effects is rare. To develop a mechanistic understanding of population dynamics in highly social species, we need to investigate how processes within groups, processes linking groups, and external drivers act and interact in space and time to produce observed patterns. Here, I consider these issues as they relate to meerkats, Suricata suricatta, obligate cooperative breeders that inhabit southern Africa. I use mathematical and statistical models, in conjunction with long-term data from a wild meerkat population, to explore population dynamics, group dynamics, group demography, Allee effects, and territory dynamics in this species. I start out by examining broad-scale patterns, and then examine some of the constituent processes. In Chapter Two, I assess the ability of phenomenological models, lacking explicit group structure, to describe population dynamics in meerkats, and I assess potential population-level Allee effects. I detect no Allee effect and conclude that explicit consideration of population structure will be key to understanding the mechanisms behind population dynamics in cooperatively breeding species. In Chapter Three, I focus on annual group-level dynamics. Using phenomenological population models, modified to incorporate environmental conditions and potential Allee effects, I first investigate overall patterns of group dynamics and find support for only conventional density dependence that increases after years of low rainfall. To explain the patterns, I examine demographic rates and assess their contributions to overall group dynamics. While per-capita meerkat mortality is subject to an Allee effect, it contributes relatively little to observed variation, and other (conventionally density dependent) demographic rates – especially emigration – govern group dynamics. In Chapter Four, I investigate group dynamics in more detail. I model demographic rates in different sex, age, and dominance classes on short timescales. Using these to build predictive and individual-based simulation models of group dynamics, I examine the demographic mechanisms responsible for declines in group size after dry years. Results reveal the delayed effect of environmental conditions, partially mediated by group structure. In Chapter Five, I explore meerkat territorial patterns. Using mechanistic home-range models, I examine group interactions, habitat selection, territory formation, and territory movement. I use meerkat data to test proposed improvements to these models, and I use the model results to start building a picture of spatial processes in meerkat population dynamics, laying the groundwork for future research. This thesis highlights the role of environment and social structure in characterizing population dynamics. I discuss the implications of my findings for the population dynamics of cooperative breeders and for population dynamics generally, noting the importance of sub-populations in drawing conclusions about socially complex systems