Population collapse to extinction: the catastrophic combination of parasitism and Allee effect

Hunting cooperation in a prey-predator model with maturation delay

We investigate the dynamics of a prey-predator model with cooperative hunting among specialist predators and maturation delay in predator growth. First, we consider a model without delay and explore the effect of hunting time on the coexistence of predator and their prey. When the hunting time is long enough and the cooperation rate among predators is weak, prey and predator species tend to coexist. Furthermore, we observe the occurrences of a series of bifurcations that depend on the cooperation rate and the hunting time. Second, we introduce a maturation delay for predator growth and analyse its impact on the system's dynamics. We find that as the delay becomes larger, predator species become more likely to go extinct, as the long maturation delay hinders the growth of the predator population. Our numerical exploration reveals that the delay causes shifts in both the bifurcation curves and bifurcation thresholds of the non-delayed system.

A model for voles interference in cultivated orchards

We consider a dynamical system involving seven populations to model the presence of voles in a cultivated orchard. The plant population is stratified by age (three groups) and by health status (being damaged or not). The last equation models the voles with a modified logistic equation with Allee effect, where the modification takes into account the disturbance provided by the human activity on the orchard. Both an analytical investigation and numerical simulations on a case study are presented. The latter support the observed differences in the literature, in terms of number of voles, between cultivated and uncultivated fields.

Can infectious diseases eradicate host species? The effect of infection-age structure

It is a fundamental question in mathematical epidemiology whether deadly infectious diseases only lead to a mere decline of their host populations or whether they can cause their complete disappearance. Upper density-dependent incidences do not lead to host extinction in simple, deterministic SI or SIS (susceptible-infectious) epidemic models. Infection-age structure is introduced into SIS models because of the biological accuracy offered by considering arbitrarily distributed infectious periods. In an SIS model with infection-age structure, survival of the susceptible host population is established for incidences that depend on the infection-age density in a general way. This confirms previous host persistence results without infection-age for incidence functions that are not generalizations of frequency-dependent transmission. For certain power incidences, hosts persist if some infected individuals leave the infected class and become susceptible again and the return rate dominates the infection-age dependent infectivity in a sufficient way. The hosts may be driven into extinction by the infectious disease if there is no return into the susceptible class at all.

Bifurcation analysis of the predator-prey model with the Allee effect in the predator

The use of predator–prey models in theoretical ecology has a long history, and the model equations have largely evolved since the original Lotka–Volterra system towards more realistic descriptions of the processes of predation, reproduction and mortality. One important aspect is the recognition of the fact that the growth of a population can be subject to an Allee effect, where the per capita growth rate increases with the population density. Including an Allee effect has been shown to fundamentally change predator–prey dynamics and strongly impact species persistence, but previous studies mostly focused on scenarios of an Allee effect in the prey population. Here we explore a predator–prey model with an ecologically important case of the Allee effect in the predator population where it occurs in the numerical response of predator without affecting its functional response. Biologically, this can result from various scenarios such as a lack of mating partners, sperm limitation and cooperative breeding mechanisms, among others. Unlike previous studies, we consider here a generic mathematical formulation of the Allee effect without specifying a concrete parameterisation of the functional form, and analyse the possible local bifurcations in the system. Further, we explore the global bifurcation structure of the model and its possible dynamical regimes for three different concrete parameterisations of the Allee effect. The model possesses a complex bifurcation structure: there can be multiple coexistence states including two stable limit cycles. Inclusion of the Allee effect in the predator generally has a destabilising effect on the coexistence equilibrium. We also show that regardless of the parametrisation of the Allee effect, enrichment of the environment will eventually result in extinction of the predator population.

The impact of individual variation on abrupt collapses in mutualistic networks

Individual variation is central to species involved in complex interactions with others in an ecological system. Such ecological systems could exhibit tipping points in response to changes in the environment, consequently leading to abrupt transitions to alternative, often less desirable states. However, little is known about how individual trait variation could influence the timing and occurrence of abrupt transitions. Using 101 empirical mutualistic networks, I model the eco‐evolutionary dynamics of such networks in response to gradual changes in strength of co‐evolutionary interactions. Results indicated that individual variation facilitates the timing of transition in such networks, albeit slightly. In addition, individual variation significantly increases the occurrence of large abrupt transitions. Furthermore, topological network features also positively influence the occurrence of such abrupt transitions. These findings argue for understanding tipping points using an eco‐evolutionary perspective to better forecast abrupt transitions in ecological systems.

Determining Optimal Stock Density of Punjab Urial (<i>Ovis vignei punjabiensis</i>) in Captivity for Breeding, Population Growth and Reintroduction Potential

BACKGROUND AND OBJECTIVE

The Punjab urial (Ovis vignei punjabiensis) is an endangered wild sheep of Pakistan, raised in captivity with the aim of re-introduction. To date, no information is available about population trends of this species in captivity. The current study was conducted with the aim to evaluate the population trend to better guide captive breeding for improved productivity and conservation value.

MATERIALS AND METHODS

Annual population data recorded and maintained by the Wildlife Department Khyber Pakhtunkhwa, Pakistan, were used. The data were compiled and analyzed in Microsoft Excel 2010 for determining growth rates and package Growthcurver in R-version 3.5.1 was used to produce a graphical representation of the population trend.

RESULTS

The overall average annual population growth rate was rN = 0.22. Results revealed a fast initial growth rate with an average value of rN = 0.4 per year. Birth rates of bN = 0.45 for the first nine years were considerably higher than the death rates dN = 0.22 and the population increased with exponential growth. In the subsequent year, very high mortalities rates (dN = 1.2), likely attributed to the clumping of the population, resulted in the collapse of the population, leaving it in a state of unstable equilibrium.

CONCLUSION

Results support the evaluation of management data to reveal carrying capacity in captive populations, to guide and inform appropriate release of surplus animals into natural habitats.

Do fatal infectious diseases eradicate host species?

In simple SI epidemic and endemic models, three classes of incidence functions are identified for their potential to be associated with host extinction: weakly upper density-dependent incidences are never associated with host extinction. Power incidences that depend on the number of susceptibles and infectives by powers strictly between 0 and 1 are associated with initial-constellation-dependent host extinction for all parameter values. Homogeneous incidences, of which frequency-dependent incidence is a very particular case, and power incidences are associated with global host extinction for certain parameter constellations and with host survival for others. Laboratory infection experiments with salamander larvae are equally well fitted by power incidences and certain upper density-dependent incidences such as the negative binomial incidence and do not rule out homogeneous incidences such as an asymmetric frequency-dependent incidence either.

Impacts of infection avoidance for populations affected by sexually transmitted infections

Sexually transmitted infections are ubiquitous in nature and affect many populations. The key process for their transmission is mating, usually preceded by mate choice. Susceptible individuals may avoid mating with infected individuals to prevent infection provided it is recognizable. We show that accounting for infection avoidance significantly alters host population dynamics. We observe bistability between the disease-free and endemic or disease-induced extinction equilibria, significant abrupt reduction in the host population size and disease-induced host extinction. From the population persistence perspective, the best strategy is either not to avoid mating with the infected individuals, to prevent disease-induced host extinction, or to completely avoid mating with the infected individuals, to prevent pathogen invasion. Increasing sterilization efficiency of the infection leads to lower population sizes and reduced effect of mating avoidance. We also find that the disease-free state is more often attained by populations with strong polyandry, whereas a high-density endemic state is more often observed for populations with strong polygyny, suggesting that polygamy rather than monogamy may be promoted in denser host populations.

Host-pathogen dynamics under sterilizing pathogens and fecundity-longevity trade-off in hosts

Infectious diseases are known to regulate population dynamics, an observation that underlies the use of pathogens as control agents of unwanted populations. Sterilizing rather than lethal pathogens are often suggested so as to avoid unnecessary suffering of the infected hosts. Until recently, models used to assess plausibility of pathogens as potential pest control agents have not included a possibility that reduced fecundity of the infected individuals may save their energy expenditure on reproduction and thus increase their longevity relative to the susceptible ones. Here, we develop a model of host-pathogen interaction that builds on this idea. We analyze the model for a variety of infection transmission functions, revealing that the indirect effect of sterilizing pathogens on mortality of the infected hosts, mediated by a fecundity-longevity trade-off, may cause hosts at endemic equilibria to attain densities higher than when there is no effect of pathogens on host mortality. On the other hand, an opposite outcome occurs when the fecundity-longevity trade-off is concave or when the degree of fecundity reduction by the pathogen is high enough. This points to a possibility that using sterilizing pathogens as agents of pest control may actually be less effective than previously thought, the more so since we also suggest that if sexual selection acts on the host species then the presence of sterilizing pathogens may even enhance host densities above the levels achieved without infection.

Diseased Social Predators

Social predators benefit from cooperation in the form of increased hunting success, but may be at higher risk of disease infection due to living in groups. Here, we use mathematical modeling to investigate the impact of disease transmission on the population dynamics benefits provided by group hunting. We consider a predator-prey model with foraging facilitation that can induce strong Allee effects in the predators. We extend this model by an infectious disease spreading horizontally and vertically in the predator population. The model is a system of three nonlinear differential equations. We analyze the equilibrium points and their stability as well as one- and two-parameter bifurcations. Our results show that weakly cooperating predators go unconditionally extinct for highly transmissible diseases. By contrast, if cooperation is strong enough, the social behavior mediates conditional predator persistence. The system is bistable, such that small predator populations are driven extinct by the disease or a lack of prey, and large predator populations survive because of their cooperation even though they would be doomed to extinction in the absence of group hunting. We identify a critical cooperation level that is needed to avoid the possibility of unconditional predator extinction. We also investigate how transmissibility and cooperation affect the stability of predator-prey dynamics. The introduction of parasites may be fatal for small populations of social predators that decline for other reasons. For invasive predators that cooperate strongly, biocontrol by releasing parasites alone may not be sufficient.

Sexually transmitted infections and mate-finding Allee effects

Infectious diseases can seriously impact dynamics of their host species. In this study, we model and analyze an interaction between a sexually transmitted infection and its animal host population affected by a mate-finding Allee effect. Since mating drives both host reproduction and infection transmission, the Allee effect shapes the transmission rate of the infection which we show takes a saturating form. Our model combining sexually transmitted infections with the mate-finding Allee effect in the host produces quite rich dynamics, including oscillations, several multistability regimes, and infection-induced host extinction. However, many of these complex patterns are restricted to a relatively narrow parameter range. We find that the host extinction occurs at intermediate levels of infection virulence, as well as for Allee effect strengths much lower than when the infection is absent. In both cases, a sequence of events comprising destabilization of an endemic equilibrium, growth of oscillation amplitude, and a heteroclinic bifurcation forms an underlying mechanism. We apply our model to the feline immunodeficiency virus (FIV) in domestic cats.

Demographic and Component Allee Effects in Southern Lake Superior Gray Wolves

Recovering populations of carnivores suffering Allee effects risk extinction because positive population growth requires a minimum number of cooperating individuals. Conservationists seldom consider these issues in planning for carnivore recovery because of data limitations, but ignoring Allee effects could lead to overly optimistic predictions for growth and underestimates of extinction risk. We used Bayesian splines to document a demographic Allee effect in the time series of gray wolf (Canis lupus) population counts (1980–2011) in the southern Lake Superior region (SLS, Wisconsin and the upper peninsula of Michigan, USA) in each of four measures of population growth. We estimated that the population crossed the Allee threshold at roughly 20 wolves in four to five packs. Maximum per-capita population growth occurred in the mid-1990s when there were approximately 135 wolves in the SLS population. To infer mechanisms behind the demographic Allee effect, we evaluated a potential component Allee effect using an individual-based spatially explicit model for gray wolves in the SLS region. Our simulations varied the perception neighborhoods for mate-finding and the mean dispersal distances of wolves. Simulation of wolves with long-distance dispersals and reduced perception neighborhoods were most likely to go extinct or experience Allee effects. These phenomena likely restricted population growth in early years of SLS wolf population recovery.

Dynamics of SI epidemic with a demographic Allee effect

In this paper, we present an extended SI model of Hilker et al. (2009). In the presented model the birth rate and the death rate are both modeled as quadratic polynomials. This approach provides ample opportunity for taking into account the major contributors to an Allee effect and effectively captures species' differential susceptibility to the Allee effects. It is shown that, the behaviors (persistence or extinction) of the model solutions are characterized by the two essential threshold parameters λ0 and λ1 of the transmissibility λ and a threshold quantity μ(∗) of the disease pathogenicity μ. If λ<λ0, the model is bistable and a disease cannot invade from arbitrarily small introductions into the host population at the carrying capacity, while it persists when λ>λ0 and μ<μ(∗). When λ>λ1 and μ>μ(∗), the disease derives the host population to extinction with origin as the only global attractor. For the special cases of the model, verifiable conditions for host population persistence (with or without infected individuals) and host extinction are derived. Interestingly, we show that if the values of the parameters α and β of the extended model are restricted, then the two models are similar. Numerical simulations show how the parameter β affects the dynamics of the model with respect to the host population persistence and extinction.

Dynamics of SI models with both horizontal and vertical transmissions as well as Allee effects

A general SI (Susceptible-Infected) epidemic system of host-parasite interactions operating under Allee effects, horizontal and/or vertical transmission, and where infected individuals experience pathogen-induced reductions in reproductive ability, is introduced. The initial focus of this study is on the analyses of the dynamics of density-dependent and frequency-dependent effects on SI models (SI-DD and SI-FD). The analyses identify conditions involving horizontal and vertical transmitted reproductive numbers, namely those used to characterize and contrast SI-FD and SI-DD dynamics. Conditions that lead to disease-driven extinction, or disease-free dynamics, or susceptible-free dynamics, or endemic disease patterns are identified. The SI-DD system supports richer dynamics including limit cycles while the SI-FD model only supports equilibrium dynamics. SI models under "small" horizontal transmission rates may result in disease-free dynamics. SI models under with and inefficient reproductive infectious class may lead to disease-driven extinction scenarios. The SI-DD model supports stable periodic solutions that emerge from an unstable equilibrium provided that either the Allee threshold and/or the disease transmission rate is large; or when the disease has limited influence on the infectives growth rate; and/or when disease-induced mortality is low. Host-parasite systems where diffusion or migration of local populations manage to destabilize them are examples of what is known as diffusive instability. The exploration of SI-dynamics in the presence of dispersal brings up the question of whether or not diffusive instability is a possible outcome. Here, we briefly look at such possibility within two-patch coupled SI-DD and SI-FD systems. It is shown that relative high levels of asymmetry, two modes of transmission, frequency dependence, and Allee effects are capable of supporting diffusive instability.

A simple epidemiological model for populations in the wild with Allee effects and disease-modified fitness

The study of the dynamics of human infectious disease using deterministic models is typically carried out under the assumption that a critical mass of individuals is available and involved in the transmission process. However, in the study of animal disease dynamics where demographic considerations often play a significant role, this assumption must be weakened. Models of the dynamics of animal populations often naturally assume that the presence of a minimal number of individuals is essential to avoid extinction. In the ecological literature, this a priori requirement is commonly incorporated as an Allee effect. The focus here is on the study disease dynamics under the assumption that a critical mass of susceptible individuals is required to guarantee the population's survival. Specifically, the emphasis is on the study of the role of an Allee effect on a Susceptible-Infectious (SI) model where the possibility that susceptible and infected individuals reproduce, with the S-class the best fit. It is further assumed that infected individuals loose some of their ability to compete for resources, the cost imposed by the disease. These features are set in motion in as simple model as possible. They turn out to lead to a rich set of dynamical outcomes. This toy model supports the possibility of multi-stability (hysteresis), saddle node and Hopf bifurcations, and catastrophic events (disease-induced extinction). The analyses provide a full picture of the system under disease-free dynamics including disease-induced extinction and proceed to identify required conditions for disease persistence. We conclude that increases in (i) the maximum birth rate of a species, or (ii) in the relative reproductive ability of infected individuals, or (iii) in the competitive ability of a infected individuals at low density levels, or in (iv) the per-capita death rate (including disease-induced) of infected individuals, can stabilize the system (resulting in disease persistence). We further conclude that increases in (a) the Allee effect threshold, or (b) in disease transmission rates, or in (c) the competitive ability of infected individuals at high density levels, can destabilize the system, possibly leading to the eventual collapse of the population. The results obtained from the analyses of this toy model highlight the significant role that factors like an Allee effect may play on the survival and persistence of animal populations. Scientists involved in biological conservation and pest management or interested in finding sustainability solutions, may find these results of this study compelling enough to suggest additional focused research on the role of disease in the regulation and persistence of animal populations. The risk faced by endangered species may turn out to be a lot higher than initially thought.

Multiscale analysis of compartment models with dispersal

The characterization of the population dynamics of animal populations and dispersal provides the underlying setting of this article. Novel results emerge from our exploration of the role of disease in this context. We focus on the study of the impact of dispersal on the dynamics of populations that account for (a) induced Allee effects; (b) disease dynamics; and (c) spatial heterogeneity, using deterministic and stochastic models. Specifically, the models incorporate disease-driven effects on the individuals' competitive ability to acquire resources as well as on their ability to move or reproduce. The results bring to the forefront the role of initial conditions and patch quality as well as 'topological' structure or connectivity landscape structure (the physical space where individuals move, reproduce, get sick, die, or compete for resources). The emphasis is placed on the dynamics of populations when disease is an important selective force. This article surveys the appropriate literature while including original research.

Fatal disease and demographic Allee effect: population persistence and extinction

If a healthy stable host population at the disease-free equilibrium is subject to the Allee effect, can a small number of infected individuals with a fatal disease cause the host population to go extinct? That is, does the Allee effect matter at high densities? To answer this question, we use a susceptible-infected epidemic model to obtain model parameters that lead to host population persistence (with or without infected individuals) and to host extinction. We prove that the presence of an Allee effect in host demographics matters even at large population densities. We show that a small perturbation to the disease-free equilibrium can eventually lead to host population extinction. In addition, we prove that additional deaths due to a fatal infectious disease effectively increase the Allee threshold of the host population demographics.