Consumer-Resource Dynamics: Quantity, Quality, and Allocation

CITATION: Getz, W. M. & Owen-Smith, N. 2011. Consumer-resource dynamics : quantity, quality, and allocation. PLoS ONE, 6(1): e14539, doi:10.1371/journal.pone.0014539.The original publication is available at http://journals.plos.org/plosoneBackground: The dominant paradigm for modeling the complexities of interacting populations and food webs is a system of coupled ordinary differential equations in which the state of each species, population, or functional trophic group is represented by an aggregated numbers-density or biomass-density variable. Here, using the metaphysiological approach to model consumer-resource interactions, we formulate a two-state paradigm that represents each population or group in a food web in terms of both its quantity and quality. Methodology and Principal Findings: The formulation includes an allocation function controlling the relative proportion of extracted resources to increasing quantity versus elevating quality. Since lower quality individuals senesce more rapidly than higher quality individuals, an optimal allocation proportion exists and we derive an expression for how this proportion depends on population parameters that determine the senescence rate, the per-capita mortality rate, and the effects of these rates on the dynamics of the quality variable. We demonstrate that oscillations do not arise in our model from quantity-quality interactions alone, but require consumer-resource interactions across trophic levels that can be stabilized through judicious resource allocation strategies. Analysis and simulations provide compelling arguments for the necessity of populations to evolve quality-related dynamics in the form of maternal effects, storage or other appropriate structures. They also indicate that resource allocation switching between investments in abundance versus quality provide a powerful mechanism for promoting the stability of consumer-resource interactions in seasonally forcing environments. Conclusions/Significance: Our simulations show that physiological inefficiencies associated with this switching can be favored by selection due to the diminished exposure of inefficient consumers to strong oscillations associated with the wellknown paradox of enrichment. Also our results demonstrate how allocation switching can explain observed growth patterns in experimental microbial cultures and discuss how our formulation can address questions that cannot be answered using the quantity-only paradigms that currently predominate. © 2011 Getz, Owen-Smith.http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0014539Publisher's versio

STOCHASTIC DELAY DIFFERENTIAL EQUATIONS WITH APPLICATIONS IN ECOLOGY AND EPIDEMICS

Mathematical modeling with delay differential equations (DDEs) is widely used for analysis and predictions in various areas of life sciences, such as population dynamics, epidemiology, immunology, physiology, and neural networks. The memory or time-delays, in these models, are related to the duration of certain hidden processes like the stages of the life cycle, the time between infection of a cell and the production of new viruses, the duration of the infectious period, the immune period, and so on. In ordinary differential equations (ODEs), the unknown state and its derivatives are evaluated at the same time instant. In DDEs, however, the evolution of the system at a certain time instant depends on the past history/memory. Introduction of such time-delays in a differential model significantly improves the dynamics of the model and enriches the complexity of the system. Moreover, natural phenomena counter an environmental noise and usually do not follow deterministic laws strictly but oscillate randomly about some average values, so that the population density never attains a fixed value with the advancement of time. Accordingly, stochastic delay differential equations (SDDEs) models play a prominent role in many application areas including biology, epidemiology and population dynamics, mostly because they can offer a more sophisticated insight through physical phenomena than their deterministic counterparts do. The SDDEs can be regarded as a generalization of stochastic differential equations (SDEs) and DDEs.This dissertation, consists of eight Chapters, is concerned with qualitative and quantitative features of deterministic and stochastic delay differential equations with applications in ecology and epidemics. The local and global stabilities of the steady states and Hopf bifurcations with respect of interesting parameters of such models are investigated. The impact of incorporating time-delays and random noise in such class of differential equations for different types of predator-prey systems and infectious diseases is studied. Numerical simulations, using suitable and reliable numerical schemes, are provided to show the effectiveness of the obtained theoretical results.Chapter 1 provides a brief overview about the topic and shows significance of the study. Chapter 2, is devoted to investigate the qualitative behaviours (through local and global stability of the steady states) of DDEs with predator-prey systems in case of hunting cooperation on predators. Chapter 3 deals with the dynamics of DDEs, of multiple time-delays, of two-prey one-predator system, where the growth of both preys populations subject to Allee effects, with a direct competition between the two-prey species having a common predator. A Lyapunov functional is deducted to investigate the global stability of positive interior equilibrium. Chapter 4, studies the dynamics of stochastic DDEs for predator-prey system with hunting cooperation in predators. Existence and uniqueness of global positive solution and stochastically ultimate boundedness are investigated. Some sufficient conditions for persistence and extinction, using Lyapunov functional, are obtained. Chapter 5 is devoted to investigate Stochastic DDEs of three-species predator prey system with cooperation among prey species. Sufficient conditions of existence and uniqueness of an ergodic stationary distribution of the positive solution to the model are established, by constructing a suitable Lyapunov functional. Chapter 6 deals with stochastic epidemic SIRC model with time-delay for spread of COVID-19 among population. The basic reproduction number ℛs0 for the stochastic model which is smaller than ℛ0 of the corresponding deterministic model is deduced. Sufficient conditions that guarantee the existence of a unique ergodic stationary distribution, using the stochastic Lyapunov functional, and conditions for the extinction of the disease are obtained. In Chapter 7, some numerical schemes for SDDEs are discussed. Convergence and consistency of such schemes are investigated. Chapter 8 summaries the main finding and future directions of research. The main findings, theoretically and numerically, show that time-delays and random noise have a significant impact in the dynamics of ecological and biological systems. They also have an important role in ecological balance and environmental stability of living organisms. A small scale of white noise can promote the survival of population; While large noises can lead to extinction of the population, this would not happen in the deterministic systems without noises. Also, white noise plays an important part in controlling the spread of the disease; When the white noise is relatively large, the infectious diseases will become extinct; Re-infection and periodic outbreaks can also occur due to the time-delay in the transmission terms

Dynamical Models of Biology and Medicine

Mathematical and computational modeling approaches in biological and medical research are experiencing rapid growth globally. This Special Issue Book intends to scratch the surface of this exciting phenomenon. The subject areas covered involve general mathematical methods and their applications in biology and medicine, with an emphasis on work related to mathematical and computational modeling of the complex dynamics observed in biological and medical research. Fourteen rigorously reviewed papers were included in this Special Issue. These papers cover several timely topics relating to classical population biology, fundamental biology, and modern medicine. While the authors of these papers dealt with very different modeling questions, they were all motivated by specific applications in biology and medicine and employed innovative mathematical and computational methods to study the complex dynamics of their models. We hope that these papers detail case studies that will inspire many additional mathematical modeling efforts in biology and medicin

Complex population dynamics in microbial systems

The study of spatial and temporal population dynamics has a long history in ecology, going back to the beginning of the 1900´s. Both intrinsic and extrinsic mechanisms are involved in determining the temporal and spatial occurrences of populations and species. Different dynamic patterns result from the strength and the interplay of the two mechanisms. The fact that in-trinsic driven population dynamics are woven together with extrinsic, often stochastic dynamics makes analyses of intrinsic mechanisms difficult and led to a controversial discussion about the relevance in nature. However, there is a gap between results from mathematical modelling showing the occurrence and meaning of intrinsically driven dynamics, and empirical proves. Recently, laboratory experiments under clearly defined and controlled conditions were shown to be a suitable tool to study intrinsic, deterministic population dynam-ics. Deterministic chaos is one type of dynamic behaviour exhibited by a change in one or more intrinsic parameters beside extinction, damped oscil-lations, and stable limit cycle. Most discussed is the relevance of chaotic be-haviour in population dynamics, due to the fact that empirical evidence is lim-ited to a simple one-species system. Furthermore, chaotic fluctuations are thought to lead to extinction of a population, because chaotic dynamics can obtain very small population sizes, even more vulnerable when mixed together with stochastic events. The question, if chaos occurs in the real world and under which circumstances chaos may be found in nature, is still open. Clearly defined laboratory experiments were established to analyse intrinsically driven dynamics in a multi-species system. Different dynamic behaviours were found in chemostat experiments with a two-prey-one-predator system of a bacterivorous ciliate as the predator and two bacteria strains as the prey organisms. The different population dynamics - extinction, damped oscillations, stable limit cycles and chaos - were triggered by a change in the dilution rate of the chemostat system and verified by calculations of the corresponding Lyapunov exponents. Therewith, chaos was shown in an experimental three-species system for the first time. The different dynamics in the microbial food web revealed a surprisingly short transition (4-7 days) to a different dynamic behaviour when the dilution rate as the control parameter was changed. All dynamics persisted in experiments when different local populations with different dynamics (chemostats with different dilution rates) were coupled. Experiments showed that the dynamic behaviours of the coupled populations were only triggered by the demographic parameter � in this case the dilution rate - and reacted independent of the constant inflow of organisms from populations with different dynamics. Here, we were able to shed more light on the question about the relevance of chaos in the real world. In conclusion spatio-temporal chaos might be more common in nature than generally assumed. Microbial communities with fast reproduction rates might be favoured candidates to show chaos and other complex dynamics in nature. Intrinsically driven dynamics might be persistent when perturbated by a constant fluctuating inflow of organisms and might lead to the establishment of chaos in habitats with constant flows (e.g. aquatic organisms in rivers and oceanic currents, and water drainage to groundwater). The fast transition to a different dynamic behaviour after a change in a control parameter shows how distinct intrinsic driven processes might be. A reason why chaotic dynamics in nature are not observed might be due too the large sampling intervals in most field studies

Chaos to Permanence-Through Control Theory

Work by Cushing et al. \cite{Cushing} and Kot et al. \cite{Kot} demonstrate that chaotic behavior does occur in biological systems. We demonstrate that chaotic behavior can enable the survival/thriving of the species involved in a system. We adopt the concepts of persistence/permanence as measures of survival/thriving of the species \cite{EVG}. We utilize present chaotic behavior and a control algorithm based on \cite{Vincent97,Vincent2001} to push a non-permanent system into permanence. The algorithm uses the chaotic orbits present in the system to obtain the desired state. We apply the algorithm to a Lotka-Volterra type two-prey, one-predator model from \cite{Harvesting}, a ratio-dependent one-prey, two-predator model from \cite{EVG} and a simple prey-specialist predator-generalist predator (for ex: plant-insect pest-spider) interaction model \cite{Upad} and demonstrate its effectiveness in taking advantage of chaotic behavior to achieve a desirable state for all species involved

Chaos to Permanence - Through Control Theory

Work by Cushing et al. [18] and Kot et al. [60] demonstrate that chaotic behavior does occur in biological systems. We demonstrate that chaotic behavior can enable the survival/thriving of the species involved in a system. We adopt the concepts of persistence/permanence as measures of survival/thriving of the species [35]. We utilize present chaotic behavior and a control algorithm based on [66, 72] to push a non-permanent system into permanence. The algorithm uses the chaotic orbits present in the system to obtain the desired state. We apply the algorithm to a Lotka-Volterra type two-prey, one-predator model from [30], a ratio-dependent one-prey, two-predator model from [35] and a simple prey-specialist predator-generalist predator (for ex: plant-insect pest-spider) interaction model [67] and demonstrate its effectiveness in taking advantage of chaotic behavior to achieve a desirable state for all species involved

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