Cyclic prey evolution with cannibalistic predators

The impact of harvesting on the evolutionary dynamics of prey species in a prey-predator systems

Matsuda and Abrams (Theor Popul Biol 45(1):76-91, 1994) initiated the exploration of self-extinction in species through evolution, focusing on the advantageous position of mutants near the extinction boundary in a prey-predator system with evolving foraging traits. Previous models lacked theoretical investigation into the long-term effects of harvesting. In our model, we introduce constant-effort prey and predator harvesting, along with individual logistic growth of predators. The model reveals two distinct evolutionary outcomes: (i) Evolutionary suicide, marked by a saddle-node bifurcation, where prey extinction results from the invasion of a lower forager mutant; and (ii) Evolutionary reversal, characterized by a subcritical Hopf bifurcation, leading to cyclic prey evolution. Employing an innovative approach based on Gröbner basis computation, we identify various bifurcation manifolds, including fold, transcritical, cusp, Hopf, and Bogdanov-Takens bifurcations. These contrasting scenarios emerge from variations in harvesting parameters while keeping other factors constant, rendering the model an intriguing subject of study.

Timescale separation and models of symbiosis: state space reduction, multiple attractors and initialization

Dynamic Energy Budget models relate whole organism processes such as growth, reproduction and mortality to suborganismal metabolic processes. Much of their potential derives from extensions of the formalism to describe the exchange of metabolic products between organisms or organs within a single organism, for example the mutualism between corals and their symbionts. Without model simplification, such models are at risk of becoming parameter-rich and hence impractical. One natural simplification is to assume that some metabolic processes act on 'fast' timescales relative to others. A common strategy for formulating such models is to assume that 'fast' processes equilibrate immediately, while 'slow' processes are described by ordinary differential equations. This strategy can bring a subtlety with it. What if there are multiple, interdependent fast processes that have multiple equilibria, so that additional information is needed to unambiguously specify the model dynamics? This situation can easily arise in contexts where an organism or community can persist in a 'healthy' or an 'unhealthy' state with abrupt transitions between states possible. To approach this issue, we offer the following: (a) a method to unambiguously complete implicitly defined models by adding hypothetical 'fast' state variables; (b) an approach for minimizing the number of additional state variables in such models, which can simplify the numerical analysis and give insights into the model dynamics; and (c) some implications of the new approach that are of practical importance for model dynamics, e.g. on the bistability of flux dynamics and the effect of different initialization choices on model outcomes. To demonstrate those principles, we use a simplified model for root-shoot dynamics of plants and a related model for the interactions between corals and endosymbiotic algae that describes coral bleaching and recovery.

Coevolution of the reckless prey and the patient predator

The war of attrition in game theory is a model of a stand-off situation between two opponents where the winner is determined by its persistence. We model a stand-off between a predator and a prey when the prey is hiding and the predator is waiting for the prey to come out from its refuge, or when the two are locked in a situation of mutual threat of injury or even death. The stand-off is resolved when the predator gives up or when the prey tries to escape. Instead of using the asymmetric war of attrition, we embed the stand-off as an integral part of the predator-prey model of Rosenzweig and MacArthur derived from first principles. We apply this model to study the coevolution of the giving-up rates of the prey and the predator, using the adaptive dynamics approach. We find that the long term evolutionary process leads to three qualitatively different scenarios: the predator gives up immediately, while the prey never gives up; the predator never gives up, while the prey adopts any giving-up rate greater than or equal to a given positive threshold value; the predator goes extinct. We observe that some results are the same as for the asymmetric war of attrition, but others are quite different.

The evolution of the irreversible transition from a free-swimming state to an immobile sessile state in aquatic invertebrates modelled in a chemostat

To better understand the environmental factors and ecological processes underlying the evolution of the irreversible transition from a free-swimming state to an immobile sessile state as seen in many aquatic invertebrates, we study the adaptive dynamics of the settling rate of a hypothetical microorganism onto the wall of a chemostat. The two states, floating or settled, differ in their nutrient ingestion, reproduction and death rate. We consider three different settling mechanisms involving competition for space on the wall: (i) purely exploitative competition where free-swimming individuals settle in vacant space only, (ii) mixed exploitative and interference competition where individuals attempt to settle in any place but fail and die if the space is already occupied, and (iii) mixed exploitative and interference competition, but now settling in occupied space is successful and the former occupant dies. In the simplified environment of the chemostat, the input concentration of nutrients and the dilution rate of the tank are the main environmental control variables. Using the theory of adaptive dynamics, we find that the settling mechanisms and environmental control variables have qualitatively different effects on the evolution of the settling rate in terms of the direction of evolution as well as on species diversity. In the case of purely exploitative competition a small change in the settings of the environmental control variables can lead to an abrupt reversal of the direction of evolution, while in the case of mixed exploitative and interference competition the effect is gradual. For all three settling mechanisms, periodic fluctuations in the nutrient input open the possibility of evolutionary branching leading to the long-term coexistence of an intermediate and an infinitely high settling rates (in the case of low-frequency fluctuations), and an intermediate and a zero settling rates (in the case of high-frequency fluctuations).

Ecological and evolutionary consequences of predator-prey role reversal: Allee effect and catastrophic predator extinction

In many terrestrial, marine, and freshwater predator-prey communities, young predators can be vulnerable to attacks by large prey. Frequent prey counter-attacks may hinder the persistence of predators. Despite the commonness of such role reversals in nature, they have rarely been addressed in evolutionary modelling. To understand how role reversals affect ecological and evolutionary dynamics of a predator-prey community, we derived an ecological model from individual-level processes using ordinary differential equations. The model reveals complex ecological dynamics, with possible bistability between alternative coexistence states and an Allee effect for the predators. We find that when prey counter-attacks are frequent, cannibalism is necessary for predator persistence. Using numerical analysis, we also find that a sudden ecological shift from coexistence to predator extinction can occur through several catastrophic bifurcations, including 'saddle-node', 'homoclinic', and 'subcritical Hopf'. The analysis of single-species evolution reveals that predator selection towards increasing or decreasing cannibalism triggers a catastrophic shift towards an extinction state of the predators. Such an evolutionary extinction of the predators may also be caused by prey selection towards increasing foraging activity because it facilitates encounters with vulnerable, young predators. The analysis of predator-prey coevolution further demonstrates that predator's catastrophic extinction becomes an even more likely outcome than in single-species evolution. Our results suggest that when young predators are vulnerable to prey attacks, a sudden extinction of the predators may be more common than currently understood.

Resident-invader dynamics of similar strategies in fluctuating environments

We study resident-invader dynamics in fluctuating environments when the invader and the resident have close but distinct strategies. First we focus on a class of continuous-time models of unstructured populations of multi-dimensional strategies, which incorporates environmental feedback and environmental stochasticity. Then we generalize our results to a class of structured population models. We classify the generic population dynamical outcomes of an invasion event when the resident population in a given environment is non-growing on the long-run and stochastically persistent. Our approach is based on the series expansion of a model with respect to the small strategy difference, and on the analysis of a stochastic fast-slow system induced by time-scale separation. Theoretical and numerical analyses show that the total size of the resident and invader population varies stochastically and dramatically in time, while the relative size of the invader population changes slowly and asymptotically in time. Thereby the classification is based on the asymptotic behavior of the relative population size, and which is shown to be fully determined by invasion criteria (i.e., without having to study the full generic dynamical system). Our results extend and generalize previous results for a stable resident equilibrium (particularly, Geritz in J Math Biol 50(1):67–82, 2005; Dercole and Geritz in J Theor Biol 394:231-254, 2016) to non-equilibrium resident population dynamics as well as resident dynamics with stochastic (or deterministic) drivers.