Spreading Speed, Traveling Waves, and Minimal Domain Size in\ud Impulsive Reaction-diffusion Models

How growth, mortality, and dispersal in a species affect the species’ spread and persistence constitutes a central problem in spatial ecology. We propose impulsive reaction-diffusion equation models for species with distinct reproductive and dispersal stages. These models can describe a seasonal birth pulse plus nonlinear mortality and dispersal throughout the year. Alternatively they can describe seasonal harvesting, plus nonlinear birth and mortality as well as dispersal throughout the year. The population dynamics in the seasonal pulse is described by a discrete map that gives the density of the populationat the end stage as a possibly nonmonotone function of the density of the population at the beginning of the stage. The dynamics in the dispersal stage is governed by a nonlinear reaction-diffusion equation in a bounded or unbounded domain. We develop a spatially explicit theoretical framework that links species vital rates (mortality or fecundity) and dispersal characteristics with species’ spreading speeds, traveling wave speeds, as well as and minimal domain size for species persistence. We provide an explicit formula for the spreading speed in terms of model parameters, and show that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions. We also determine an explicit formula for the minimal domain size using model parameters. Our results show how the diffusion coefficient, and the combination of discrete- and continuous-time growth and mortality determine the spread and persistence dynamics of the population in a wide variety of ecological scenarios. Numerical simulations are presented to demonstrate the theoretical results

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

Traveling wavefronts of a prey–predator diffusion system with stage-structure and harvesting

AbstractFrom a biological point of view, we consider a prey–predator-type free diffusion fishery model with stage-structure and harvesting. First, we study the stability of the nonnegative constant equilibria. In particular, the effect of harvesting on the stability of equilibria is discussed and supported with numerical simulation. Then, employing the upper and lower solution method, we show that when the wave speed is large enough there exists a traveling wavefront connecting the zero solution to the positive equilibrium of the system. Numerical simulation is also carried out to illustrate the main result

Predator persistence through variability of resource productivity in Tritrophic systems

The trophic structure of species communities depends on the energy transfer between trophic levels. Primary productivity varies strongly through time, challenging the persistence of species at higher trophic levels. Yet resource variability has mostly been studied in systems with only one or two trophic levels. We test the effect of variability in resource productivity in a tritrophic model system including a resource, a size-structured consumer, and a size-specific predator. The model complies with fundamental principles of mass conservation and the body-size dependence of individual-level energetics and predator-prey interactions. Surprisingly, we find that resource variability may promote predator persistence. The positive effect of variability on the predator arises through periods with starvation mortality of juvenile prey, which reduces the intraspecific competition in the prey population. With increasing variability in productivity and starvation mortality in the juvenile prey, the prey availability increases in the size range preferred by the predator. The positive effect of prey mortality on the trophic transfer efficiency depends on the biologically realistic consideration of body size–dependent and food-dependent functions for growth and reproduction in our model. Our findings show that variability may promote the trophic transfer efficiency, indicating that environmental variability may sustain species at higher trophic levels in natural ecosystems

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

Impact of irrigation on poverty and environment in Ethiopia. Draft Proceeding of the Symposium and Exhibition held at Ghion Hotel, Addis Ababa, Ethiopia 27th -29th November, 2007

Poverty, Crop management, Irrigated farming, Rainfed farming, Irrigation systems, Food security, Water harvesting, Institutions, Environmental effects, Public health, Malaria, GIS, Remote sensing, Crop Production/Industries, Environmental Economics and Policy, Farm Management, Food Consumption/Nutrition/Food Safety, Food Security and Poverty, Health Economics and Policy, Institutional and Behavioral Economics, Resource /Energy Economics and Policy,

A computational investigation of seasonally forced disease dynamics

In recent years there has been a great increase in work on epidemiological modelling, driven partly by the increase in the availability and power of computers, but also by the desire to improve standards of public and animal health. Through modelling, understanding of the mechanisms of previous epidemics can be gained, and the lessons learnt applied to make predictions about future epidemics, or emerging diseases. The standard SIR model is in some sense quite a simplistic model, and can lack realism. One solution to this problem is to increase the complexity of the model, or to perform full scale simulation—an experiment in silico. This thesis, however, takes a different approach and makes an in depth analysis of one small improvement to the model: the replacement of a constant birth rate with a birth pulse. This more accurately describes the seasonal birth patterns observed in many animal populations. The combination of the nonlinearities of the SIR model and the strong seasonal forcing provided by the birth pulse necessitate the use of numerical methods. The model shows complex multi annual cycles of epidemics and even chaos for shorter infectious periods. The robustness of these results are proven with respect to a wide range or perturbations: in phase space, in the shape and temporal extent of the birth pulse and in the underlying model to which the pulsing is applied. To complement the numerics, analytic methods are used to gain further understanding of the dynamics in particular areas of the chosen parameter space where the numerics can be challenging. Three approximations are presented, one to investigate very small levels of forcing, and two covering short infectious periods.EThOS - Electronic Theses Online ServiceEngineering and Physical Sciences Research Council (EPSRC)GBUnited Kingdo

Physiological Bases of Plant Shrinkage and Its Demographic Implications

Perennial plants are plastic organisms and can vary greatly in size within and between years. While much attention has been paid to the factors that maximize plant growth, plant shrinkage has been bluntly overlooked. Here we study anatomy, physiology, demography and comparative biology to understand how often plants shrink, mechanisms by which plants may internally regulate their architecture to shrink, and shrinkage’s demographic implications. We explored belowground resource uptake, hydraulic transport and demographic contributions of shrinkage of individuals of Cryptantha flava (Boraginaceae) in the Great Basin desert. We used a database with \u3e700 species to explore shrinkage’s frequency and ecological consequences by examining demographic parameters such as population growth rate, longevity and reproductive output for species with and without shrinkage. We show that C. flava can forage the heterogeneous desert space by uptaking via some roots but not others, and that individuals become sectored with age (i.e., individual roots transport resources only to specific canopy regions). The switch in hydraulic design with ontogeny may explain why large individuals are more likely to shrink after droughts than juveniles. Perturbation analyses demonstrated that large individuals contribute very little to the population growth rate (λ) and that growth to large classes can negatively affect λ in C. flava. The comparative analyses indicated that shrinkage correlates positively with lifespan, survival, fecundity and growth. Finally, we found a significant number of candidate herbaceous species where shrinkage might be under selection because in them shrinking increases λ. We have linked anatomical traits, physiological mechanisms and population dynamics to offer a mechanistic explanation of how and why plants shrink. We argue that sectoriality in Cryptantha flava may allow individuals to minimize respiratory demands during harsh conditions by shedding sectors. Thus shrinkage, being preferable over mortality, is a survival mechanism that helps individuals buffer stochasticity. The finding that large individuals of C. flava contribute very little to λ in comparison to medium-sized individuals and that positive shrinkage vital-rate sensitivities occur frequently in the plant kingdom, together with recent demographic analytical developments, make exploring plant shrinkage a very promising venue of ecological and evolutionary research

Applications with Discrete and Continuous Models: Harvesting and Contact Tracing

Harvest plays an important role in management decisions, from fisheries to pest control. Discrete models enable us to explore the importance of timing of management decisions including the order of events of particular actions. We derive novel mechanistic models featuring explicit within season harvest timing and level. Our models feature explicit discrete density independent birth pulses, continuous density dependent mortality, and density independent harvest level at a within season harvest time. We explore optimization of within-season harvest level and timing through optimal control of these population models. With a fixed harvest level, harvest timing is taken as the control. Then with fixed timing, the harvest level is implemented as the control. Finally, both harvest timing and level are used as controls. We maximize an objective functional which includes management goals of maximizing yield, maximizing stock, and minimizing costs associated with both harvest intensity and harvest timing. The 2014-2016 West African outbreak of Ebola Virus Disease (EVD) was the largest and most deadly to date. Contact tracing, following up those who may have been infected through contact with an infected individual to prevent secondary spread, plays a vital role in controlling such outbreaks. However, there were many complications and challenges to contact tracing efforts during the 2014-2016 outbreak. We present a system of ordinary differential equations to model contact tracing in Sierra Leone during the outbreak. Using data on cumulative cases and deaths we estimate most of the parameters in our model. We include the novel features of counting the total number of people being traced and tying this directly to the number of tracers doing this work. We explore the role contact tracing played in eventually ending the outbreak and examine the potential impact of improved contact tracing on the death toll