Evolution of pathogens with cross-immunity in response to healthcare interventions.

Cross-immunity, as an evolutionary driver, can contribute to pathogen evolution, particularly pathogen diversity. Healthcare interventions aimed at reducing disease severity or transmission are commonly used to control diseases and can also induce pathogen evolution. Understanding pathogen evolution in the context of cross-immunity and healthcare interventions is crucial for infection control. This study starts by modelling cross-immunity, the extent of which is determined by strain traits and host characteristics. Given that all hosts have the same characteristics, full cross-immunity between residents and mutants occurs when mutation step sizes are small enough. Cross-immunity can be partial when the step size is large. The presence of partial cross-immunity reduces pathogen load and shortens the infectious period inside hosts, reducing transmission between hosts and improving host population survival and recovery. This study focuses on how pathogens evolve through small and large mutational steps and how healthcare interventions affect pathogen evolution. Using the theory of adaptive dynamics, we found that when mutational steps are small (only full cross-immunity is present), pathogen diversity cannot occur because it maximises the basic reproduction number. This results in intermediate values for both pathogen growth and clearance rates. However, when large mutational steps are allowed (with full and partial cross-immunity present), pathogens can evolve into multiple strains and induce pathogen diversity. The study also shows that different healthcare interventions can have varying effects on pathogen evolution. Generally, low levels of intervention are more likely to induce strain diversity, while high levels are more likely to result in strain reduction.

Modelling and optimising healthcare interventions in a model with explicit within- and between-host dynamics.

Given an endemic infectious disease and a budget, how do we optimally allocate interventions to control the disease? This paper shows that the optimal strategy varies depending on the budget, the type of intervention, the trajectory of pathogen load, and the objective. Using a model with explicit within- and between-host dynamics, we model isolation, supportive treatment, and specific treatment. Isolation and supportive treatment affect the transmission coefficient and the disease-induced mortality rate, respectively, in the between-host dynamics. Specific treatment affects the clearance rate of pathogens in the within-host dynamics. We study the optimisation of the three interventions for various budget levels via evaluating isolation and supportive treatment at the population level and specific treatment at both the population and individual levels. At the population level, we consider the risk of transmission, the burden of illness, and the survival probability, and to that end, we choose the population-level infection rate, the population density of infected individuals, and the total disease-induced mortality rate as objective functions. At the individual level, we consider the length of infection and the pathogen load, and to that end, we choose the maximum infection-age and the maximum pathogen load as objective functions. The objective is to minimise these functions through varying two variables that refer to when the intervention starts and when it stops for an infected individual and also indicate what kind of individuals can get the intervention from the population perspective. We find that the optimal strategy of isolation is to isolate individuals with a higher pathogen load, given a lower budget. The optimal strategy of supportive treatment can be the same as isolation or simply no treatment. The optimal strategy of specific treatment is complicated, and it can be to treat individuals with pathogen loads above a particular level until they recover or until the pathogens can decrease when treatment stops, or it can be another scenario.

Virulence management: Closing the feedback loop between healthcare interventions and virulence evolution.

We study the evolution of virulence of an endemic pathogen in response to healthcare interventions which affect host recovery and pathogen transmission. By anticipating the evolutionary response of the pathogen we may develop effective long-term management strategies for controlling the impact of the endemic on the society. To that end, we use standard Adaptive Dynamics techniques in an SIS model. The recovery rate and the transmission rate, both of which can be affected by healthcare interventions, are used as evolutionary control variables. The effect of interventions may be density-independent (self-help based on healthcare instructions) or density-dependent (when assistance of a healthcare worker is required). We consider the evolutionary response of the pathogen both to abrupt changes and to gradual changes in the level of healthcare intervention. Healthcare intervention is optimised for three alternative objectives: minimisation of virulence, minimisation of the probability that an infected individual dies of the disease, and total eradication of the endemic. We find that the optimal strategy may depend on the objective. High levels of healthcare intervention may eradicate the pathogen, but this option may not be available for budgetary reasons or otherwise. Counterintuitively, to minimise virulence, one should keep healthcare interventions at a minimum, while to minimise the probability for an infected individual to die of the disease, both low and high levels of healthcare intervention suffice. Changes in the level of healthcare intervention should be implemented fast (not gradually) in order to avoid sudden changes in pathogen evolution and the possible emergence of multiple simultaneously coexisting pathogen strains.

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).

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.

Correction to: Adaptive dynamics of saturated polymorphisms.

In the original publication, Proposition 4 is mistaken.

Coevolution of cannibalistic predators and timid prey: evolutionary cycling and branching.

We investigate the coevolution of cannibalistic predators and timid prey, which seek refuge upon detecting a predator. To understand how the species affect each other's evolution, we derived the ecological model from individual-level processes using ordinary differential equations. The ecological dynamics exhibit bistability between equilibrium and periodic attractors, which may disappear through catastrophic bifurcations. Using the critical function analysis of adaptive dynamics, we classify general trade-offs between cannibalism and prey capture that produce different evolutionary outcomes. The evolutionary analysis reveals several ways in which cannibalism emerges as a response to timidity of the prey. The long-term coevolution either attains a singularity, or becomes cyclic through two mechanisms: genetical cycles through Hopf bifurcation of the singularity, or ecogenetical cycles involving abrupt switching between ecological attractors. Further diversification of cannibalism occurs through evolutionary branching, which is predicted to be delayed when simultaneous prey evolution is necessary for the singularity's attainability. We conclude that predator-prey coevolution produces a variety of outcomes, in which evolutionary cycles are commonplace.

Cyclic prey evolution with cannibalistic predators.

We investigate the evolution of timidity in a prey species whose predator has cannibalistic tendencies. The ecological model is derived from individual-level processes, in which the prey seeks refuge after detecting a predator, and the predator cannibalises on the conspecific juveniles. Bifurcation analysis of the model reveals ecological bistability between equilibrium and periodic attractors. Using the framework of adaptive dynamics, we classify ten qualitatively different evolutionary scenarios induced by the ecological bistability. These scenarios include ecological attractor switching through catastrophic bifurcations, which can reverse the direction of evolution. We show that such reversals often result in evolutionary cycling of the level of timidity. In the absence of cannibalism, the model never exhibits ecological bistability nor evolutionary cycling. We conclude that cannibalistic predator behaviour can completely change both the ecological dynamics and the evolution of prey.

Frequency dependence 3.0: an attempt at codifying the evolutionary ecology perspective.

The fitness concept and perforce the definition of frequency independent fitnesses from population genetics is closely tied to discrete time population models with non-overlapping generations. Evolutionary ecologists generally focus on trait evolution through repeated mutant substitutions in populations with complicated life histories. This goes with using the per capita invasion speed of mutants as their fitness. In this paper we develop a concept of frequency independence that attempts to capture the practical use of the term by ecologists, which although inspired by population genetics rarely fits its strict definition. We propose to call the invasion fitnesses of an eco-evolutionary model frequency independent when the phenotypes can be ranked by competitive strength, measured by who can invade whom. This is equivalent to the absence of weak priority effects, protected dimorphisms and rock-scissor-paper configurations. Our concept differs from that of Heino et al. (TREE 13:367-370, 1998) in that it is based only on the signs of the invasion fitnesses, whereas Heino et al. based their definitions on the structure of the feedback environment, summarising the effect of all direct and indirect interactions between individuals on fitness. As it turns out, according to our new definition an eco-evolutionary model has frequency independent fitnesses if and only if the effect of the feedback environment on the fitness signs can be summarised by a single scalar with monotonic effect. This may be compared with Heino et al.'s concept of trivial frequency dependence defined by the environmental feedback influencing fitness, and not just its sign, in a scalar manner, without any monotonicity restriction. As it turns out, absence of the latter restriction leaves room for rock-scissor-paper configurations. Since in 'realistic' (as opposed to toy) models frequency independence is exceedingly rare, we also define a concept of weak frequency dependence, which can be interpreted intuitively as almost frequency independence, and analyse in which sense and to what extent the restrictions on the potential model outcomes of the frequency independent case stay intact for models with weak frequency dependence.

Unfolding the resident-invader dynamics of similar strategies.

We investigate the competition between two groups of similar agents in the restricted, but classical context of unstructured populations varying in continuous time in an isolated, homogeneous, and constant abiotic environment. Individual behavioral and phenotypic traits are quantified by one-dimensional strategies and intra- as well as inter-specific interactions are described in the vicinity of a stationary regime. Some known results are revisited: invasion by a new strategy generically implies the substitution of the former resident; and resident-invader coexistence is possible close to singular strategies-the stationary points of the invasion fitness-and is generically protected-each of the two competing groups can invade the other. An (almost known) old conjecture is shown true: competition close to a singular strategy is "essentially Lotka-Volterra"-dominance of one strategy, protected coexistence at an intermediate equilibrium, and mutual exclusion are the generic outcomes. And the unfolding of the competition scenarios is completed with the analysis of three degenerate singular strategies-characterized by vanishing second-order fitness derivatives-near which resident-invader coexistence can be unprotected. Our approach is based on the series expansion of a generic demographic model, w.r.t. the small strategy difference between the two competing groups, and on known results on time-scale separation and bifurcation theories. The analysis is carried out up to third order and is extendable to any order. For each order, explicit genericity conditions under which higher orders can be neglected are derived and, interestingly, they are known prior to invasion. An important result is that degeneracies up to third-order are required to have more than one stable way of coexistence. Such degeneracies can be due to particular symmetries in the model formulation, and breaking the genericity conditions provides a direct way to draw biological interpretations. The developed body of theory is exemplified on a model for the evolution of cannibalism and on Lotka-Volterra competition models.

Adaptive dynamics of saturated polymorphisms.

We study the joint adaptive dynamics of n scalar-valued strategies in ecosystems where n is the maximum number of coexisting strategies permitted by the (generalized) competitive exclusion principle. The adaptive dynamics of such saturated systems exhibits special characteristics, which we first demonstrate in a simple example of a host-pathogen-predator model. The main part of the paper characterizes the adaptive dynamics of saturated polymorphisms in general. In order to investigate convergence stability, we give a new sufficient condition for absolute stability of an arbitrary (not necessarily saturated) polymorphic singularity and show that saturated evolutionarily stable polymorphisms satisfy it. For the case [Formula: see text], we also introduce a method to construct different pairwise invasibility plots of the monomorphic population without changing the selection gradients of the saturated dimorphism.

Mutual invadability near evolutionarily singular strategies for multivariate traits, with special reference to the strongly convergence stable case.

Over the last two decades evolutionary branching has emerged as a possible mathematical paradigm for explaining the origination of phenotypic diversity. Although branching is well understood for one-dimensional trait spaces, a similarly detailed understanding for higher dimensional trait spaces is sadly lacking. This note aims at getting a research program of the ground leading to such an understanding. In particular, we show that, as long as the evolutionary trajectory stays within the reign of the local quadratic approximation of the fitness function, any initial small scale polymorphism around an attracting invadable evolutionarily singular strategy (ess) will evolve towards a dimorphism. That is, provided the trajectory does not pass the boundary of the domain of dimorphic coexistence and falls back to monomorphism (after which it moves again towards the singular strategy and from there on to a small scale polymorphism, etc.). To reach these results we analyze in some detail the behavior of the solutions of the coupled Lande-equations purportedly satisfied by the phenotypic clusters of a quasi-n-morphism, and give a precise characterisation of the local geometry of the set D in trait space squared harbouring protected dimorphisms. Intriguingly, in higher dimensional trait spaces an attracting invadable ess needs not connect to D. However, for the practically important subset of strongly attracting ess-es (i.e., ess-es that robustly locally attract the monomorphic evolutionary dynamics for all possible non-degenerate mutational or genetic covariance matrices) invadability implies that the ess does connect to D, just as in 1-dimensional trait spaces. Another matter is that in principle there exists the possibility that the dimorphic evolutionary trajectory reverts to monomorphism still within the reign of the local quadratic approximation for the invasion fitnesses. Such locally unsustainable branching cannot occur in 1- and 2-dimensional trait spaces, but can do so in higher dimensional ones. For the latter trait spaces we give a condition excluding locally unsustainable branching which is far stricter than the one of strong convergence, yet holds good for a relevant collection of published models. It remains an open problem whether locally unsustainable branching can occur around general strongly attracting invadable ess-es.

The DeAngelis-Beddington functional response and the evolution of timidity of the prey.

We study the evolution of "timidity" of the prey (i.e., its readiness to seek refuge) in a predator-prey model with the DeAngelis-Beddington functional response. Using the theory of adaptive dynamics, we find that a predator-prey population at equilibrium always favours less timidity. Low levels of timidity, however, may destabilise the population and lead to cycles. Large-amplitude cycles favour a positive level of timidity, but if such cycles do not occur, timidity will evolve all the way to zero, where the prey no longer responds to the predator by seeking refuge, and in which case the DeAngelis-Beddington functional response has become identical to the Holling type-II functional response.

Evolution of pathogen virulence under selective predation: a construction method to find eco-evolutionary cycles.

We investigate eco-evolutionary cycles in the joint dynamics of pathogen virulence and predator population density when hosts carrying virulent infections are exposed to increased risk of predation. We introduce a new technique to find trade-off functions under which the model exhibits limit cycles; this technique provides a constructive proof that the system is able to generate limit cycles, and can be applied to other eco-evolutionary models as well. We also study a concrete example to confirm that eco-evolutionary cycles occur in a significant part of the parameter space and to briefly explore other evolutionary outcomes in the same model.

On the evolution of the timing of reproduction with non-equilibrium resident dynamics.

We study the evolution of an individual's reproductive strategy in a mechanistic modeling framework. We assume that the total number of juveniles one adult individual can produce is a finite constant, and we study how this number should be distributed during the season, given the types of inter-individual interactions and mortality processes included in the model. The evolution of the timing of reproduction in this modeling framework has already been studied earlier in the case of equilibrium resident dynamics, but we generalize the situation to also fluctuating population dynamics. We find that, as in the equilibrium case, the presence or absence of inter-juvenile aggression affects the functional form of the evolutionarily stable reproductive strategy. If an ESS exists, it can have an absolutely continuous part only if inter-juvenile aggression is included in the model. If inter-juvenile aggression is not included in the model, an ESS can have no continuous parts, and only Dirac measures are possible.

Evolution of density-dependent dispersal in a structured metapopulation.

We study the evolution of density-dependent dispersal in a structured metapopulation subject to local catastrophes that eradicate local populations. To this end we use the theory of structured metapopulation dynamics and the theory of adaptive dynamics. The set of evolutionarily possible dispersal functions (i.e., emigration rates as a function of the local population density) is derived mechanistically from an underlying resource-consumer model. The local resource dynamics is of a flow-culture type and consumers leave a local population with a constant probability per unit of time kappa when searching for resources but not when handling resources (i.e., eating and digesting). The time an individual spends searching (as opposed to handling) depends on the local resource density, which in turn depends on the local consumer density, and so the average per capita emigration rate depends on the local consumer density as well. The derived emigration rates are sigmoid functions of local consumer population density. The parameters of the local resource-consumer dynamics are subject to evolution. In particular, we find that there exists a unique evolutionarily stable and attracting dispersal rate kappa( *) for searching consumers. The kappa( *) increases with local resource productivity and decreases with resource decay rate. The kappa( *) also increases with the survival probability during dispersal, but as a function of the catastrophe rate it reaches a maximum before dropping off to zero again.

Evolutionary branching and long-term coexistence of cycling predators: critical function analysis.

It is well known that two predators with different functional responses can coexist on one prey when the system exhibits nonequilibrium dynamics. In this paper, we investigate under which conditions such coexistence is evolutionarily stable, and whether the two predators may evolve from a single ancestor via evolutionary branching. We assume that predator strategies differ in handling time, and hence in the shape of their Holling type II functional response. Longer handling times are costly in terms of lost foraging time, but allow the predator to extract more nutrients from the prey and therefore to produce more offspring per consumed prey. In the analysis, we apply a new method to accommodate arbitrary trade-off functions between handling time and offspring production. Contrary to previous results obtained assuming a particular trade-off [Kisdi, E. and Liu, S., 2006. J. Evol. Biol. 19, 49-58], we find that evolutionary branching of handling time is possible, although it does not appear to be very likely and can be excluded for a class of trade-offs. Evolutionarily stable coexistence of two predators occurs under less restrictive conditions, which are always satisfied when the trade-off function has two strongly concave parts connected by a convex piece.