Invasion Threshold Research Articles

Analysis of Monod type food chain chemostat with k-times’ periodically pulsed input

In this paper, we introduce and study a model of a predator-prey system with Monod type functional response under periodic pulsed chemostat conditions, which contains with predator, prey, and periodically pulsed substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations. Furthermore, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the impulsive system shows two kinds of bifurcations, whose are period-doubling and period-halfing.

A fungal growth model fitted to carbon‐limited dynamics of Rhizoctonia solani

Here, a quasi-steady-state approximation was used to simplify a mathematical model for fungal growth in carbon-limiting systems, and this was fitted to growth dynamics of the soil-borne plant pathogen and saprotroph Rhizoctonia solani. The model identified a criterion for invasion into carbon-limited environments with two characteristics driving fungal growth, namely the carbon decomposition rate and a measure of carbon use efficiency. The dynamics of fungal spread through a population of sites with either low (0.0074 mg) or high (0.016 mg) carbon content were well described by the simplified model with faster colonization for the carbon-rich environment. Rhizoctonia solani responded to a lower carbon availability by increasing the carbon use efficiency and the carbon decomposition rate following colonization. The results are discussed in relation to fungal invasion thresholds in terms of carbon nutrition.

DYNAMIC COMPLEXITIES OF A CHEMOSTAT MODEL WITH PULSED INPUT AND WASHOUT AT DIFFERENT TIMES

In this paper, we consider a predator–prey chemostat model with ratio-dependent Monod type functional response and periodic input and washout at different fixed times. We obtain an exact periodic solution with substrate and prey. The stability analysis for this periodic solutions yields an invasion threshold for the period of pulses of the predator. When the impulsive period is more than the threshold, there are periodic oscillations in the substrate, prey, and predator. If the impulsive period further increases, the system undergoes a complex dynamic process. By analyzing bifurcation diagrams, we can see that the impulsive system shows two kinds of bifurcation, which are period-doubling and period-halving.

Bifurcation and chaos of Tessiet type food chain chemostat with periodically varying substrate

In this paper, we introduce and study a model of Tessiet type food chain chemostat with periodically varying substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey and predator. Furthermore, we numerically simulate a model with sinusoidal dilution rate, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the periodic system shows two kinds of bifurcations, whose are period-doubling and period-halving.

Invasion thresholds and the evolution of nonequilibrium virulence.

The enterprise of virulence management attempts to predict how social practices and other factors affect the evolution of parasite virulence. These predictions are often based on parasite optima or evolutionary equilibria derived from models of host-parasite dynamics. Yet even when such models accurately capture the parasite optima, newly invading parasites will typically not be at their optima. Here we show that parasite invasion of a host population can occur despite highly nonoptimal virulence. Fitness improvements soon after invasion may proceed through many steps with wide changes in virulence, because fitness depends on transmission as well as virulence, and transmission improvements can overwhelm nonoptimal virulence. This process is highly sensitive to mutation supply and the strength of selection. Importantly, the same invasion principle applies to the evolution of established parasites, whenever mutants arise that overcome host immunity/resistance. A host population may consequently experience repeated invasions of new parasite variants and possible large shifts in virulence as it evolves in an arms race with the parasite. An experimental study of phage lysis time and examples of mammalian viruses matching some of these characteristics are reviewed.

Mating preference in the invasion of growth enhanced fish

Fish with a transgene for growth hormone grow faster than the wild type and may have an advantage in sexual selection due to their larger size and earlier maturation. The cost in these genetically modified organisms (GMOs) is a lower viability of their offspring. The Trojan gene effect is a hypothesis that predicts that the release of such fish in nature can lead to an invasion by GMOs but ultimately decrease population size to extinction. We modelled GMO invasion with Mendelian inheritance of two alleles in one locus and the resulting mating and population dynamics of wild, GMO and hybrid genotypes. Invasion was attempted over a range of initial densities, representing scenarios from accidental escape to large‐scale deliberate introduction of the transgenic genotype. Our results show that invasion strongly depends on hybrid fitness, requiring only a low initial density when GMOs and hybrids are preferred in mating. Preference against hybrids results in an invasion threshold, above which mating between GMOs are sufficiently frequent for invasion to take place. GMO invasion may decrease population size, but contrary to earlier studies on the Trojan gene effect, extinctions do not occur. This is due to the lower viability of GMOs being balanced by the decreased number of competitors reducing the effects of density dependence. The results emphasize the importance of initial density, hybrid fitness and density dependence when considering invasion through hybridization.

Analysis of a Beddington–DeAngelis food chain chemostat with periodically varying substrate

In this paper, we introduce and study a model of a Beddington–DeAngelis type food chain chemostat with periodically varying substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey and predator. Furthermore, we numerically simulate a model with sinusoidal input, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the periodic system shows two kinds of bifurcations, whose are period-doubling and period-halfing.

Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: Theory and simulations

The spatial structure of populations is a key element in the understanding of the large-scale spreading of epidemics. Motivated by the recent empirical evidence on the heterogeneous properties of transportation and commuting patterns among urban areas, we present a thorough analysis of the behavior of infectious diseases in metapopulation models characterized by heterogeneous connectivity and mobility patterns. We derive the basic reaction–diffusion equations describing the metapopulation system at the mechanistic level and derive an early stage dynamics approximation for the subpopulation invasion dynamics. The analytical description uses a homogeneous assumption on degree block variables that allows us to take into account arbitrary degree distribution of the metapopulation network. We show that along with the usual single population epidemic threshold the metapopulation network exhibits a global threshold for the subpopulation invasion. We find an explicit analytic expression for the invasion threshold that determines the minimum number of individuals traveling among subpopulations in order to have the infection of a macroscopic number of subpopulations. The invasion threshold is a function of factors such as the basic reproductive number, the infectious period and the mobility process and it is found to decrease for increasing network heterogeneity. We provide extensive mechanistic numerical Monte Carlo simulations that recover the analytical finding in a wide range of metapopulation network connectivity patterns. The results can be useful in the understanding of recent data driven computational approaches to disease spreading in large transportation networks and the effect of containment measures such as travel restrictions.

Analysis of a Beddington–DeAngelis food chain chemostat with periodically varying dilution rate

A model of a Beddington–DeAngelis type food chain chemostat with periodically varying dilution rate is considered. The system consists of one predator, one prey and limiting substrate. The periodic solutions of the subsystem with substrate and prey, which are the boundary periodic solutions of the system, are obtained. The stability analysis of the boundary periodic solution yields a predator invasion threshold. Above this threshold, there are periodic oscillations in the system. Furthermore, a model with sinusoidal dilution rate is numerically simulated. By bifurcation diagrams with different bifurcation parameters, the periodic system shows two kinds of bifurcations, whose are period-doubling and period-halving.

Invasion Threshold in Heterogeneous Metapopulation Networks

We study the dynamics of epidemic and reaction-diffusion processes in metapopulation models with heterogeneous connectivity patterns. In susceptible-infected-removed-like processes, along with the standard local epidemic threshold, the system exhibits a global invasion threshold. We provide an explicit expression of the threshold that sets a critical value of the diffusion/mobility rate below, which the epidemic is not able to spread to a macroscopic fraction of subpopulations. The invasion threshold is found to be affected by the topological fluctuations of the metapopulation network. The results presented provide a general framework for the understanding of the effect of travel restrictions in epidemic containment.

Study of a chemostat model with Beddington–DeAngelis functional response and pulsed input and washout at different times

In the paper, we considered a food chain chemostat model with a Beddington–DeAngelis functional response of predator, with periodical input and washout occurring at different fixed times. We obtained exact periodic solutions for the model with substrate and prey only. The stability analysis for this periodic solutions yields an invasion threshold for period of pulses of the predator. When the impulsive period is greater than the threshold, there are periodic oscillations in substrate, prey, and predator. If the impulsive period is increased further, the system undergoes the complex dynamic process. By analyzing bifurcation diagrams, we can see that the impulsive system shows two kinds of bifurcations; period-doubling and period-halving.

Analysis of a Tessiet type food chain chemostat with k-times’ periodically pulsed input

In this paper, we introduce and study a Tessiet type food chain chemostat, which contains with predator, prey and k-times’ periodically pulsed substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey, and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations. Furthermore, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the impulsive system shows two kinds of bifurcations, whose are period-doubling and period-halfing. When impulsive period is small, there exists quasiperiodic oscillation in the impulsive system.

A simple model for the dynamics of a host–parasite–hyperparasite interaction

Hyperparasites can play a crucial role in the control of a host–parasite interaction if they are successfully established in the community. We investigated the specific traits of the hyperparasite and those of the release event which allow a successful regulation of primary parasite populations. This study has been motivated by the case study of chestnut– Cryphonectria parasitica–Cryphonectria Hypovirus interaction. We use a model of SIR/SIS type which assumes a limited diffusion of the parasite. Our model emphasizes the thresholds for invasion linked to the ecological specificities of both the pathogen and the hyperparasite (transmission rates and virulence) and to the initial conditions of the system (population sizes of the different categories). The predictions are consistent with data on the observed spread of the virus. “Mild” strains of the hyperparasite, characterized by a high vertical transmission rate and low virulence, are more prone to establish than “severe” strains. It also demonstrates that the horizontal transmission of the virus, which is controlled by a vegetative incompatibility system in the fungus, is not the unique constraint for the virus establishment. This study may contribute to theoretical and practical aspects of the biological control of plant diseases with a hyperparasite and to the ecology of biological invasions.

Study of Lotka-volterra food chain chemostat with periodically varying dilution rate

In this paper, we introduce and study a model of Lotka-volterra chemostat food chain chemostat with periodically varying dilution rate, which contains with predator, prey, and substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey, and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations. Furthermore, we numerically simulate a model with sinusoidal dilution rate, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the system experiences following process: periodic solution → periodic doubling cascade → chaos.

The Durability of Public Goods Changes the Dynamics and Nature of Social Dilemmas

An implicit assumption underpins basic models of the evolution of cooperation, mutualism and altruism: The benefits (or pay-offs) of cooperation and defection are defined by the current frequency or distribution of cooperators. In social dilemmas involving durable public goods (group resources that can persist in the environment–ubiquitous from microbes to humans) this assumption is violated. Here, we examine the consequences of relaxing this assumption, allowing pay-offs to depend on both current and past numbers of cooperators. We explicitly trace the dynamic of a public good created by cooperators, and define pay-offs in terms of the current public good. By raising the importance of cooperative history in determining the current fate of cooperators, durable public goods cause novel dynamics (e.g., transient increases in cooperation in Prisoner's Dilemmas, oscillations in Snowdrift Games, or shifts in invasion thresholds in Stag-hunt Games), while changes in durability can transform one game into another, by moving invasion thresholds for cooperation or conditions for coexistence with defectors. This enlarged view challenges our understanding of social cheats. For instance, groups of cooperators can do worse than groups of defectors, if they inherit fewer public goods, while a rise in defectors no longer entails a loss of social benefits, at least not in the present moment (as highlighted by concerns over environmental lags). Wherever durable public goods have yet to reach a steady state (for instance due to external perturbations), the history of cooperation will define the ongoing dynamics of cooperators.

Analysis of a Monod–Haldene type food chain chemostat with seasonally variably pulsed input and washout

In this paper, we introduce and study a model of a Monod–Haldene type food chain chemostat with seasonally variably pulsed input and washout. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations. Furthermore, bifurcation diagrams have shown that there exists complexity for the pulsed system including periodic doubling cascade, periodic halving cascade and Pitchfork bifurcations and tangent bifurcations.

Analysis of a Monod–Haldene type food chain chemostat with periodically varying substrate

In this paper, we introduce and study a model of a Monod–Haldene type food chain chemostat with periodically varying substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey and predator. Furthermore, we numerically simulate a model with sinusoidal input, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the periodic system shows two kinds of bifurcations, whose are period-doubling and period-halfing.

Study of a Monod–Haldene type food chain chemostat with pulsed substrate

In this paper, we introduce and study a model of a Monod-Haldene type food chain chemostat with pulsed substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey, and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations. Furthermore, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the impulsive system shows two kinds of bifurcations, whose are period-doubling and period-halfing.

Analysis of competitive chemostat models with the Beddington–DeAngelis functional response and impulsive effect

In this paper, we introduce and study a competitive system with Beddington–DeAngelis type functional response in periodic pulsed chemostat conditions. We investigate the subsystem with substrate and one of the microorganisms and study the stability of the periodic solutions, which are the boundary periodic solution of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate and one of the microorganism. Further, we prove that the system is permanent if the impulsive period less than some critical value. Therefore, our results are valuable for the manufacture of products by genetically altered organisms.

Evolution and invasion dynamics of multiple infections with Wolbachia investigated using matrix based models

Endosymbiotic bacteria are often transmitted vertically from one host generation to the next via oocytes cytoplasm. The generally small number of colonizing bacteria in the oocytes leads to a bottleneck at each generation, resulting in genetic homogenization of the symbiotic population. Nevertheless, in many of the species infected by Wolbachia (maternally transmitted bacteria), individuals do sometimes simultaneously harbor several bacterial strains, owing to the fact that Wolbachia induces cytoplasmic incompatibility (CI) that maintains multiple infections. CI occurs in crosses in which the male is infected by at least one Wolbachia strain that the female lacks, and consequently it favors individuals with the greatest symbiotic diversity. CI results in death of offspring in diploid species. In haplodiploid individuals, unfertilized eggs hatch normally into males and fertilized ones, which would lead to females, either die (female mortality type: FM) or develop into males (male development type: MD). Until now, only one theoretical study, restricted to diploid species, has investigated the associations where multiple CI-inducing Wolbachia co-exist, and explored the conditions under which multiple infections can spread. The consequences of double infections on Wolbachia maintenance in host populations, and the selective pressures to which it is subjected have not yet been analysed. Here, we have re-written a model previously developed for single infection in matrix form, which allows easy extension to multiple infections and introduction of mutant strains. We show that (i) the CI type has a strong influence on invasiveness and maintenance of multiple infections; (ii) double infection lowers the invasion threshold of less competitive strains that hitch-hike with their companion strain; (iii) when multiple infections occur, as in single infections, the strains selected are those which maximize the production of infected offspring; and (iv) for the MD CI type, invasion of mutant strains can carry the whole infection to extinction.