Rapid Evolution of Resistance and Tolerance Leads to Variable Host Recoveries following Disease-Induced Declines.

AbstractRecoveries of populations that have suffered severe disease-induced declines are being observed across disparate taxa. Yet we lack theoretical understanding of the drivers and dynamics of recovery in host populations and communities impacted by infectious disease. Motivated by disease-induced declines and nascent recoveries in amphibians, we developed a model to ask the following question: How does the rapid evolution of different host defense strategies affect the transient recovery trajectories of hosts following pathogen invasion and disease-induced declines? We found that while host life history is predictably a major driver of variability in population recovery trajectories (including declines and recoveries), populations that use different host defense strategies (i.e., tolerance, avoidance resistance, and intensity-reduction resistance) experience notably different recoveries. In single-species host populations, populations evolving tolerance recovered on average four times slower than populations evolving resistance. Moreover, while populations using avoidance resistance strategies had the fastest potential recovery rates, these populations could get trapped in long transient states at low abundance prior to recovery. In contrast, the recovery of populations evolving intensity-reduction resistance strategies were more consistent across ecological contexts. Overall, host defense strategies strongly affect the transient dynamics of population recovery and may affect the ultimate fate of real populations recovering from disease-induced declines.

Overcoming the impossibility of age-balanced harvest.

In many countries, sustainability targets for managed fisheries are often expressed in terms of a fixed percentage of the carrying capacity. Despite the appeal of such a simple quantitative target, an unintended consequence may be a significant tilting of the proportions of biomass across different ages, from what they would have been under harvest-free conditions. Within the framework of a widely used age-structured model, we propose a novel quantitative definition of "age-balanced harvest" that considers the age-class composition relative to that of the unfished population. We show that achieving a perfectly age-balanced policy is impossible if we harvest any fish whatsoever. However, every non-trivial harvest policy has a special structure that favours the young. To quantify the degree of age-imbalance, we propose a cross-entropy function. We formulate an optimisation problem that aims to attain an "age-balanced steady state", subject to adequate yield. We demonstrate that near balanced harvest policies are achievable by sacrificing a small amount of yield. These findings have important implications for sustainable fisheries management by providing insights into trade-offs and harvest policy recommendations.

Technique to derive discrete population models with delayed growth.

We provide a procedure for deriving discrete population models for the size of the adult population at the beginning of each breeding cycle and assume only adult individuals reproduce. This derivation technique includes delay to account for the number of breeding cycles that a newborn individual remains immature and does not contribute to reproduction. These models include a survival probability (during the delay period) for the immature individuals, since these individuals have to survive to reach maturity and become members of, what we consider, the adult population. We discuss properties of this class of discrete delay population models and show that there is a critical delay threshold. The population goes extinct if the delay exceeds this threshold. We apply this derivation procedure to obtain two models, a Beverton-Holt adult model and a Ricker adult model and discuss the global dynamics of both models.

The Beverton-Hold model on isolated time scales.

In this work, we formulate the Beverton-Holt model on isolated time scales and extend existing results known in the discrete and quantum calculus cases. Applying a recently introduced definition of periodicity for arbitrary isolated time scales, we discuss the effects of periodicity onto a population modeled by a dynamic version of the Beverton-Holt equation. The first main theorem provides conditions for the existence of a unique ω -periodic solution that is globally asymptotically stable, which addresses the first Cushing-Henson conjecture on isolated time scales. The second main theorem concerns the generalization of the second Cushing-Henson conjecture. It investigates the effects of periodicity by deriving an upper bound for the average of the unique periodic solution. The obtained upper bound reveals a dependence on the underlying time structure, not apparent in the classical case. This work also extends existing results for the Beverton-Holt model in the discrete and quantum cases, and it complements existing conclusions on periodic time scales. This work can furthermore guide other applications of the recently introduced periodicity on isolated time scales.

Square root identities for harvested Beverton-Holt models.

We introduce the term net-proliferation rate for a class of harvested single species models, where harvest is assumed to reduce the survival probability of individuals. Following the classical maximum sustainable yield calculations, we establish relations between the proliferation and net-proliferation that are economically and sustainably favored. The resulting square root identities are analytically derived for species following the Beverton-Holt recurrence considering three levels of complexity. To discuss the generalization of the results, we compare the square root result to the optimal survival rate of the Pella-Tomlinson model. Furthermore, to test the practical relevance of the square root identities, we fit a stochastic Pella-Tomlinson model to observed Barramundi fishery data from the Southern Gulf of Carpentaria, Australia. The results show that for the estimated model parameters, the equilibrium biomass levels resulting from the MSY harvest and the square root harvest are similar, supporting the claim that the square root harvest can serve as a rule-of-thumb. This application, with its inherited model uncertainty, sparks a risk sensitivity analysis regarding the probability of populations falling below an unsustainable threshold. Characterization of such sensitivity helps in the understanding of both dangers of overfishing and potential remedies.

Derivation and Analysis of a Discrete Predator-Prey Model.

We derive a discrete predator-prey model from first principles, assuming that the prey population grows to carrying capacity in the absence of predators and that the predator population requires prey in order to grow. The proposed derivation method exploits a technique known from economics that describes the relationship between continuous and discrete compounding of bonds. We extend standard phase plane analysis by introducing the next iterate root-curve associated with the nontrivial prey nullcline. Using this curve in combination with the nullclines and direction field, we show that the prey-only equilibrium is globally asymptotic stability if the prey consumption-energy rate of the predator is below a certain threshold that implies that the maximal rate of change of the predator is negative. We also use a Lyapunov function to provide an alternative proof. If the prey consumption-energy rate is above this threshold, and hence the maximal rate of change of the predator is positive, the discrete phase plane method introduced is used to show that the coexistence equilibrium exists and solutions oscillate around it. We provide the parameter values for which the coexistence equilibrium exists and determine when it is locally asymptotically stable and when it destabilizes by means of a supercritical Neimark-Sacker bifurcation. We bound the amplitude of the closed invariant curves born from the Neimark-Sacker bifurcation as a function of the model parameters.

An alternative delayed population growth difference equation model.

We propose an alternative delayed population growth difference equation model based on a modification of the Beverton-Holt recurrence, assuming a delay only in the growth contribution that takes into account that those individuals that die during the delay, do not contribute to growth. The model introduced differs from a delayed logistic difference equation, known as the delayed Pielou or delayed Beverton-Holt model, that was formulated as a discretization of the Hutchinson model. The analysis of our delayed difference equation model identifies a critical delay threshold. If the time delay exceeds this threshold, the model predicts that the population will go extinct for all non-negative initial conditions. If the delay is below this threshold, the population survives and its size converges to a positive globally asymptotically stable equilibrium that is decreasing in size as the delay increases. We show global asymptotic stability of the positive equilibrium using two different techniques. For one set of parameter values, a contraction mapping result is applied, while the proof for the remaining set of parameter values, relies on showing that the map is eventually componentwise monotone.

Mistakes can stabilise the dynamics of rock-paper-scissors games.

A game of rock-paper-scissors is an interesting example of an interaction where none of the pure strategies strictly dominates all others, leading to a cyclic pattern. In this work, we consider an unstable version of rock-paper-scissors dynamics and allow individuals to make behavioural mistakes during the strategy execution. We show that such an assumption can break a cyclic relationship leading to a stable equilibrium emerging with only one strategy surviving. We consider two cases: completely random mistakes when individuals have no bias towards any strategy and a general form of mistakes. Then, we determine conditions for a strategy to dominate all other strategies. However, given that individuals who adopt a dominating strategy are still prone to behavioural mistakes in the observed behaviour, we may still observe extinct strategies. That is, behavioural mistakes in strategy execution stabilise evolutionary dynamics leading to an evolutionary stable and, potentially, mixed co-existence equilibrium.

A measure for intrinsic information.

We introduce an information measure that reflects the intrinsic perspective of a receiver or sender of a single symbol, who has no access to the communication channel and its source or target. The measure satisfies three desired properties-causality, specificity, intrinsicality-and is shown to be unique. Causality means that symbols must be transmitted with probability greater than chance. Specificity means that information must be transmitted by an individual symbol. Intrinsicality means that a symbol must be taken as such and cannot be decomposed into signal and noise. It follows that the intrinsic information carried by a specific symbol increases if the repertoire of symbols increases without noise (expansion) and decreases if it does so without signal (dilution). An optimal balance between expansion and dilution is relevant for systems whose elements must assess their inputs and outputs from the intrinsic perspective, such as neurons in a network.

Optimal harvesting policy for the Beverton--Holt model.

In this paper, we establish the exploitation of a single population modeled by the Beverton--Holt difference equation with periodic coefficients. We begin our investigation with the harvesting of a single autonomous population with logistic growth and show that the harvested logistic equation with periodic coefficients has a unique positive periodic solution which globally attracts all its solutions. Further, we approach the investigation of the optimal harvesting policy that maximizes the annual sustainable yield in a novel and powerful way; it serves as a foundation for the analysis of the exploitation of the discrete population model. In the second part of the paper, we formulate the harvested Beverton--Holt model and derive the unique periodic solution, which globally attracts all its solutions. We continue our investigation by optimizing the sustainable yield with respect to the harvest effort. The results differ from the optimal harvesting policy for the continuous logistic model, which suggests a separate strategy for populations modeled by the Beverton--Holt difference equation.