Numerical equilibrium analysis for structured consumer resource models.

In this paper, we present methods for a numerical equilibrium and stability analysis for models of a size structured population competing for an unstructured resource. We concentrate on cases where two model parameters are free, and thus existence boundaries for equilibria and stability boundaries can be defined in the (two-parameter) plane. We numerically trace these implicitly defined curves using alternatingly tangent prediction and Newton correction. Evaluation of the maps defining the curves involves integration over individual size and individual survival probability (and their derivatives) as functions of individual age. Such ingredients are often defined as solutions of ODE, i.e., in general only implicitly. In our case, the right-hand sides of these ODE feature discontinuities that are caused by an abrupt change of behavior at the size where juveniles are assumed to turn adult. So, we combine the numerical solution of these ODE with curve tracing methods. We have implemented the algorithms for "Daphnia consuming algae" models in C-code. The results obtained by way of this implementation are shown in the form of graphs.

Culling experiments demonstrate size-class specific biomass increases with mortality.

Size-selective mortality inevitably leads to a decrease in population density and exerts a direct negative effect on targeted size classes. But density and population size structure are also shaped by food-dependent processes, such as individual growth, maturation, and reproduction. Mortality relaxes competition and thereby alters the dynamic interplay among these processes. As shown by the recently developed size-structured theory, which can account for food-dependent individual performance, this altered interplay can lead to overcompensatory responses in size class-specific biomass, with increasing mortality. We experimentally tested this theory by subjecting laboratory fish populations to a range of size-selective mortality rates. Overall, the results were in agreement with theoretical predictions. Biomass of the juvenile size class increased above control levels at intermediate adult mortality rates and thereafter declined at high mortality rates. Juvenile biomass also increased when juveniles themselves were subjected to intermediate mortality rates. Biomass in other size classes decreased with mortality. Such biomass overcompensation can have wide-ranging implications for communities and food webs, including a high sensitivity of top predators to irreversible catastrophic collapses, the establishment of alternative stable community states, and the promotion of coexistence and biodiversity.

Stage-specific predator species help each other to persist while competing for a single prey.

Prey in natural communities are usually shared by many predator species. How predators coexist while competing for the same prey is one of the fundamental questions in ecology. Here, we show that competing predator species may not only coexist on a single prey but even help each other to persist if they specialize on different life history stages of the prey. By changing the prey size distribution, a predator species may in fact increase the amount of prey available for its competitor. Surprisingly, a predator may not be able to persist at all unless its competitor is also present. The competitor thus significantly increases the range of conditions for which a particular predator can persist. This "emergent facilitation" is a long-term, population-level effect that results from asymmetric increases in the rate of prey maturation and reproduction when predation relaxes competition among prey. Emergent facilitation explains observations of correlated increases of predators on small and large conspecific prey as well as concordance in their distribution patterns. Our results suggest that emergent facilitation may promote the occurrence of complex, stable, community food webs and that persistence of these communities could critically depend on diversity within predator guilds.

Population feedback after successful invasion leads to ecological suicide in seasonal environments.

For most consumer species, winter represents a period of harsh food conditions in addition to the physiological strain that results from the low ambient temperatures. In size-structured populations, larger-bodied individuals do better during winter as they have larger energy reserves to buffer starvation periods. In contrast, smaller-bodied individuals do better under growing conditions, as they have lower maintenance costs. We study how the interplay between size-dependent life-history processes and seasonal changes in temperature and food availability shape the long-term dynamics of a size-structured consumer population and its unstructured resource. We show that the size dependence of maintenance requirements translates into a minimum body size that is needed for surviving starvation when consumers can adapt only to a limited extent to the low food densities in winter. This size threshold can lead to population extinction because adult individuals suffer only a little during winter and hence produce large numbers of offspring. Due to population feedback on the resource and intense intra-cohort competition, newborn consumers then fail to reach the size threshold for survival. Under these conditions, small numbers of individuals can survive, increase in density, and build up a population, which will subsequently go extinct due to its feedback on the resource. High juvenile mortality may prevent this ecological suicide from occurring, as it releases resource competition among newborns and speeds up their growth. In size-structured populations, annual fluctuations in temperature and food availability may thus lead to a conflict between individual fitness and population persistence.

Size-dependent interactions inhibit coexistence in intraguild predation systems with life-history omnivory.

Growth in body size during ontogeny often results in changes in diet, leading to life-history omnivory. In addition, growth is often dependent on food density. Using a physiologically structured population model, we investigated the effects of these two aspects of individual growth in a system consisting of two size-structured populations, an omnivorous top predator and an intermediate consumer. With a single shared resource for both populations, we found that life-history omnivory decreases the likelihood of coexistence between top predator and intermediate consumer in this intraguild predation (IGP) system. This result contrasts with previous unstructured models and stage-structured models without food-dependent development. Food-dependent development and size-dependent foraging abilities of the predator resulted in a positive feedback between foraging success on the shared resource at an early life stage and foraging success on the intermediate consumer later in life. By phenomenologically incorporating this feedback in an unstructured IGP model, we show that it also demotes coexistence in this simple setting, demonstrating the robustness of the negative effect of this feedback.

On the (dis) advantages of cannibalism.

Cannibalism is an interaction between individuals that can produce counter- intuitive effects at the population level. A striking effect is that a population may persist under food conditions such that the non-cannibalistic variant is doomed to go extinct. This so-called life boat mechanism has received considerable attention. Implicitly, such studies sometimes suggest, that the life boat mechanism procures an evolutionary advantage to the cannibalistic trait. Here we compare, in the context of a size structured population model, the conditions under which the life boat mechanism works, with those that guarantee, that a cannibalistic mutant can invade successfully under the steady environmental conditions as set by a non-cannibalistic resident. We find qualitative agreement and quantitative difference. In particular, we find that a prerequisite for the life boat mechanism is, that cannibalistic mutants are successful invaders. Roughly speaking, our results show that cannibalism brings advantages to both the individuals and the population when adult food is limiting.

Impact of intraguild predation and stage structure on simple communities along a productivity gradient.

We analyze the consequences of intraguild predation and stage structure for the possible composition of a three-species community consisting of resource, consumer, and predator. Intraguild predation, a special case of omnivory, induces two major differences with traditional linear food chain models: the potential for the occurrence of two alternative stable equilibria at intermediate levels of resource productivity and the extinction of the consumer at high productivities. At low productivities, the consumer dominates, while at intermediate productivities, the predator and the consumer can coexist. The qualitative behavior of the model is robust against addition of an invulnerable size class for the consumer population and against addition of an initial, nonpredatory stage for the predator population, which means that the addition of stage structure does not change the pattern. Unless the top predator is substantially less efficient on the bottom resource, it tends to drive the intermediate species extinct over a surprisingly large range of productivities, thus making coexistence generally impossible. These theoretical results indicate that the conditions for stable food chains involving intraguild predation cannot involve strong competition for the bottommost resource.

Ontogenetic scaling of foraging rates and the dynamics of a size-structured consumer-resource model.

The ontogenetic scaling of foraging capacity strongly influences the competitive ability of differently sized individuals within a species. We develop a physiologically structured model to investigate the effect of different ontogenetic size scalings of the attack rate on the population dynamics of a consumer-resource system. The resource is assumed to reproduce continuously whereas the consumer only reproduces at discrete time instants. Depending on the ontogenetic size scaling, the model exhibited recruit-driven cycles, stable fixed point dynamics, non-recruit juvenile-driven cycles, quasiperiodic orbits, or chaotic dynamics. The kind of dynamics observed was related to the maintenance resource levels required of differently sized individuals. Stable fixed point dynamics was, besides at the persistence boundary, only observed when the minimum resource levels were similar for newborns and mature individuals. The tendency for large population fluctuations over a wide range of the parameter space was due to the consumer's pulsed reproduction. Background mortality and length of season were major determinants of cycle length. Model dynamics strongly resembled empirically observed dynamics from fish and Daphnia populations with respect to both patterns and mechanisms. The non-recruit juvenile-driven dynamics is suggested to occur in populations with size-dependent interference or preemptive competition like cicada populations.

Population level consequences of toxicological influences on individual growth and reproduction in Lumbricus rubellus (Lumbricidae, Oligochaeta).

The effects of increased environmental concentrations of copper on the population dynamics of Lumbricus rubellus are investigated. A size-structured matrix model is used to translate sublethal effects on individual growth and reproduction into their population dynamical consequences. Laboratory data on growth and reproduction under different, sublethal conditions of copper stress are used to parameterize the model. An estimate for the critical threshold concentration of copper (critical in a sense that the population growth rate at this concentration equals zero), obtained from the model analysis, agrees well with observations on field populations of L. rubellus.

Dynamics of age-structured and spatially structured predator-prey interactions: individual-based models and population-level formulations.

In this article, we investigate the spatial and temporal dynamics of predator and prey populations using an individual-based modeling approach. In our models, the individual is the fundamental unit, and the dynamics are governed by individual rules for growth, movement, reproduction, feeding, and mortality. We first establish the congruence between age-structured predator-prey population models and the corresponding individual-based population model under homogeneous spatial conditions. Given the agreement between the formalisms, we then use the individual-based model to investigate the dynamics of spatially structured predator-prey systems. In particular, we contrast the dynamics of predator-prey systems in which predators adopt either an "ambush" or a "cruising" strategy. We show that the stability of the spatially structured predator-prey system depends on the relative mobility of prey and predators and that prey mobility, in particular, has a strong effect on stability. Local density dependence in prey reproduction can quantitatively alter the asymmetrical influence of prey mobility on stability, but we show that the asymmetry exists when local density dependence is removed. We hypothesize that this asymmetrical response is due to prey "escape" in space caused by differences in rates of spread of prey and predator populations that arise because of fundamental differences between prey and predator reproduction.