Deterministic and Stochastic Analysis of a Ratio-Dependent Predator-Prey System with Delay

Recently ratio-dependent predator-prey models have become the focus of considerable attention in theoretical ecology in their own right. In this paper, we have studied the deterministic and stochastic dynamical aspects of stability of a MichaelisMenten type ratio-dependent predator-prey system that includes discrete time-delay. Computer simulations are carried out to explain the analytical findings in deterministic environment. The biological implications of our analytical and numerical findings are discussed critically

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

A Modified Holling-Tanner Model in Stochastic Environment

Recently, a modified version of the so called Holling-Tannermodel isintroduced in the ecological literature. A detailed account of the deterministic dynamicsof this model is presented. The growth rates of the prey and predator are then perturbedby Gaussian white noises to take into account the effect of fluctuating environment. Theresulting stochastic model is cultured by the technique of statistical linearization andcriteria for non-equilibrium fluctuation and stability arederived. Numerical simulationsare carried out. The implications of our analytical findingsare addressed critically

Indirect effects of primary prey population dynamics on alternative prey

We develop a theory of generalist predation showing how alternative prey species are affected by changes in both mean abundance and variability (coefficient of variation) of their predator's primary prey. The theory is motivated by the indirect effects of cyclic rodent populations on ground-breeding birds, and developed through progressive analytic simplifications of an empirically-based model. It applies nonetheless to many other systems where primary prey have fast life-histories and can become locally superabundant, which facilitates impact on alternative prey species. In contrast to classic apparent competition theory based on symmetric interactions, our results suggest that predator effects on alternative prey should generally decrease with mean primary prey abundance, and increase with primary prey variability (low to high CV) - unless predators have strong aggregative responses, in which case these results can be reversed. Approximations of models including predator dynamics (general numerical response with possible delays) confirm these results but further suggest that negative temporal correlation between predator and primary prey is harmful to alternative prey. We find in general that predator numerical responses are crucial to predict the response of ecosystems to changes in key prey species exhibiting outbreaks, and extend the apparent competition/mutualism theory to asymmetric interactions

Noise Induced Phenomena in the Dynamics of Two Competing Species

Noise through its interaction with the nonlinearity of the living systems can give rise to counter-intuitive phenomena. In this paper we shortly review noise induced effects in different ecosystems, in which two populations compete for the same resources. We also present new results on spatial patterns of two populations, while modeling real distributions of anchovies and sardines. The transient dynamics of these ecosystems are analyzed through generalized Lotka-Volterra equations in the presence of multiplicative noise, which models the interaction between the species and the environment. We find noise induced phenomena such as quasi-deterministic oscillations, stochastic resonance, noise delayed extinction, and noise induced pattern formation. In addition, our theoretical results are validated with experimental findings. Specifically the results, obtained by a coupled map lattice model, well reproduce the spatial distributions of anchovies and sardines, observed in a marine ecosystem. Moreover, the experimental dynamical behavior of two competing bacterial populations in a meat product and the probability distribution at long times of one of them are well reproduced by a stochastic microbial predictive model.Comment: 23 pages, 8 figures; to be published in Math. Model. Nat. Phenom. (2016

Drivers of population cycles in ecological systems

In this thesis, mathematical models are used to investigate potential drivers of population cycles. Population cycles are a common ecological phenomenon, yet the mechanisms underpinning these oscillations are not always known. We focus on two distinct systems, and evaluate potential causes of cyclic dynamics. In the first part of the thesis, we develop and analyse a host–pathogen model, incorporating density-dependent prophylaxis (DDP). DDP describes when individuals invest more in immunity at high population densities, due to the increased risk of becoming infected by a pathogen. The implications of this for the population dynamics of both host and pathogen are examined. We find that the delay in the onset of DDP is critical in determining whether DDP increases or decreases the likelihood of population cycles. Secondly, we focus on a particular cyclic vole population, that of Kielder Forest, Northern UK. We construct a model to test the hypothesis that the population oscillations observed in this location are caused by the interaction between the voles and the silica in the grass they consume. We extend our model by including seasonal forcing, and study the effects of this on the population dynamics.Engineering and Physical Sciences Research Council (EPSRC

Complex Dynamics in a Singular Delayed Bioeconomic Model with and without Stochastic Fluctuation

A singular delayed biological economic predator-prey system with and without stochastic fluctuation is proposed. The conditions of singularity induced bifurcation are gained, and a state feedback controller is designed to eliminate such bifurcation. Furthermore, saddle-node bifurcation is also showed. Next, the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the distribution of roots of the corresponding characteristic equation, and the hybrid control strategy is used to control the occurrence of Hopf bifurcation. In addition, some explicit formulas determining the spectral densities of the populations and harvest effort are given when the system is considered with stochastic fluctuation. Finally, numerical simulations are illustrated to verify the theoretical results