Absolute stability and dynamical stabilisation in predator-prey systems

HOW DO PARTIALLY HIDING BEHAVIOR AND COMPLETELY HIDING BEHAVIOR IMPACT INTERACTING POPULATIONS?

As the two main behaviors of prey populations in ecological systems, the partially hiding behavior (PHB) and the completely hiding behavior (CHB) play a significant role in determining the dynamics of predator–prey models. This work examines to the dynamical consequences of predator–prey systems with the PHB and the CHB. Previous research has independently studied the two behaviors, and the general conclusions are that the two behaviors can have positive and/or negative impacts on the considered population models. However, to our knowledge, no study has combined and compared the two behaviors in studying the dynamical consequences of predation interactions. Motivated by this, we investigated the dynamical consequences induced by the PHB and the CHB. From a mathematical point of view, the dynamical behaviors are studied and the corresponding sufficient conditions are given. Our findings are general and some published models are special cases of ours. From an ecological point of view, we find that the size of the ecological regions is mainly determined by the two behaviors, and which one is ecologically beneficial for the health coexistence of the interacting populations are primarily determined by the functional response and the attack coefficient of predators. Moreover, we conclude that the evolutionary and optimal choices of prey behavior (PHB or CHB) depend on the predators attack coefficient (large or small attack coefficient) and the resource level (abundant or pool resource level).

Long transients in ecology: Theory and applications

This paper discusses the recent progress in understanding the properties of transient dynamics in complex ecological systems. Predicting long-term trends as well as sudden changes and regime shifts in ecosystems dynamics is a major issue for ecology as such changes often result in population collapse and extinctions. Analysis of population dynamics has traditionally been focused on their long-term, asymptotic behavior whilst largely disregarding the effect of transients. However, there is a growing understanding that in ecosystems the asymptotic behavior is rarely seen. A big new challenge for theoretical and empirical ecology is to understand the implications of long transients. It is believed that the identification of the corresponding mechanisms along with the knowledge of scaling laws of the transient's lifetime should substantially improve the quality of long-term forecasting and crisis anticipation. Although transient dynamics have received considerable attention in physical literature, research into ecological transients is in its infancy and systematic studies are lacking. This text aims to partially bridge this gap and facilitate further progress in quantitative analysis of long transients in ecology. By revisiting and critically examining a broad variety of mathematical models used in ecological applications as well as empirical facts, we reveal several main mechanisms leading to the emergence of long transients and hence lays the basis for a unifying theory.

Revealing new dynamical patterns in a reaction-diffusion model with cyclic competition via a novel computational framework

Understanding how patterns and travelling waves form in chemical and biological reaction-diffusion models is an area which has been widely researched, yet is still experiencing fast development. Surprisingly enough, we still do not have a clear understanding about all possible types of dynamical regimes in classical reaction-diffusion models, such as Lotka-Volterra competition models with spatial dependence. In this study, we demonstrate some new types of wave propagation and pattern formation in a classical three species cyclic competition model with spatial diffusion, which have been so far missed in the literature. These new patterns are characterized by a high regularity in space, but are different from patterns previously known to exist in reaction-diffusion models, and may have important applications in improving our understanding of biological pattern formation and invasion theory. Finding these new patterns is made technically possible by using an automatic adaptive finite element method driven by a novel a posteriori error estimate which is proved to provide a reliable bound for the error of the numerical method. We demonstrate how this numerical framework allows us to easily explore the dynamical patterns in both two and three spatial dimensions.

Tumour chemotherapy strategy based on impulse control theory

Chemotherapy is a widely accepted method for tumour treatment. A medical doctor usually treats patients periodically with an amount of drug according to empirical medicine guides. From the point of view of cybernetics, this procedure is an impulse control system, where the amount and frequency of drug used can be determined analytically using the impulse control theory. In this paper, the stability of a chemotherapy treatment of a tumour is analysed applying the impulse control theory. The globally stable condition for prescription of a periodic oscillatory chemotherapeutic agent is derived. The permanence of the solution of the treatment process is verified using the Lyapunov function and the comparison theorem. Finally, we provide the values for the strength and the time interval that the chemotherapeutic agent needs to be applied such that the proposed impulse chemotherapy can eliminate the tumour cells and preserve the immune cells. The results given in the paper provide an analytical formula to guide medical doctors to choose the theoretical minimum amount of drug to treat the cancer and prevent harming the patients because of over-treating.This article is part of the themed issue 'Horizons of cybernetical physics'.

Pattern selection and hysteresis in the Rietkerk model for banded vegetation in semi-arid environments

Banded vegetation is a characteristic feature of semi-arid environments. It occurs on gentle slopes, with alternating stripes of vegetation and bare ground running parallel to the contours. A number of mathematical models have been proposed to investigate the mechanisms underlying these patterns, and how they might be affected by changes in environmental conditions. One of the most widely used models is due to Rietkerk and co-workers, and is based on a water redistribution hypothesis, with the key feedback being that the rate of rainwater infiltration into the soil is an increasing function of plant biomass. Here, for the first time, we present a detailed study of the existence and stability of pattern solutions of the Rietkerk model on slopes, using the software package wavetrain (www.ma.hw.ac.uk/wavetrain). Specifically, we calculate the region of the rainfall-migration speed parameter plane in which patterns exist, and the sub-region in which these patterns are stable as solutions of the model partial differential equations. We then perform a detailed simulation-based study of the way in which patterns evolve when the rainfall parameter is slowly varied. This reveals complex behaviour, with sudden jumps in pattern wavelength, and hysteresis; we show that these jumps occur when the contours of constant pattern wavelength leave the parameter region giving stable patterns. Finally, we extend our results to the case in which a diffusion term for surface water is added to the model equations. The parameter regions for pattern existence and stability are relatively insensitive to small or moderate levels of surface water diffusion, but larger diffusion coefficients significantly change the subdivision into stable and unstable patterns.

A mathematical biologist's guide to absolute and convective instability

Mathematical models have been highly successful at reproducing the complex spatiotemporal phenomena seen in many biological systems. However, the ability to numerically simulate such phenomena currently far outstrips detailed mathematical understanding. This paper reviews the theory of absolute and convective instability, which has the potential to redress this inbalance in some cases. In spatiotemporal systems, unstable steady states subdivide into two categories. Those that are absolutely unstable are not relevant in applications except as generators of spatial or spatiotemporal patterns, but convectively unstable steady states can occur as persistent features of solutions. The authors explain the concepts of absolute and convective instability, and also the related concepts of remnant and transient instability. They give examples of their use in explaining qualitative transitions in solution behaviour. They then describe how to distinguish different types of instability, focussing on the relatively new approach of the absolute spectrum. They also discuss the use of the theory for making quantitative predictions on how spatiotemporal solutions change with model parameters. The discussion is illustrated throughout by numerical simulations of a model for river-based predator-prey systems.