Extinction times for certain predator–prey processes

1988 ◽  
Vol 25 (03) ◽  
pp. 612-616
Author(s):  
C. J. Ridler-Rowe

Finiteness of mean extinction times for certain predator-prey models has been established by Hitchcock (1986) with the aid of a criterion of Reuter (1957). Using this criterion and a ‘minimisation' lemma this note shows that the mean extinction times tend to zero as the combined initial population of predators and prey becomes large.

1986 ◽  
Vol 23 (01) ◽  
pp. 1-13
Author(s):  
S. E. Hitchcock

Two stochastic models are developed for the predator-prey process. In each case it is shown that ultimate extinction of one of the two populations is certain to occur in finite time. For each model an exact expression is derived for the probability that the predators eventually become extinct when the prey birth rate is 0. These probabilities are used to derive power series approximations to extinction probabilities when the prey birth rate is not 0. On comparison with values obtained by numerical analysis the approximations are shown to be very satisfactory when initial population sizes and prey birth rate are all small. An approximation to the mean number of changes before extinction occurs is also obtained for one of the models.


1986 ◽  
Vol 23 (1) ◽  
pp. 1-13 ◽  
Author(s):  
S. E. Hitchcock

Two stochastic models are developed for the predator-prey process. In each case it is shown that ultimate extinction of one of the two populations is certain to occur in finite time. For each model an exact expression is derived for the probability that the predators eventually become extinct when the prey birth rate is 0. These probabilities are used to derive power series approximations to extinction probabilities when the prey birth rate is not 0. On comparison with values obtained by numerical analysis the approximations are shown to be very satisfactory when initial population sizes and prey birth rate are all small. An approximation to the mean number of changes before extinction occurs is also obtained for one of the models.


1988 ◽  
Vol 25 (3) ◽  
pp. 612-616 ◽  
Author(s):  
C. J. Ridler-Rowe

Finiteness of mean extinction times for certain predator-prey models has been established by Hitchcock (1986) with the aid of a criterion of Reuter (1957). Using this criterion and a ‘minimisation' lemma this note shows that the mean extinction times tend to zero as the combined initial population of predators and prey becomes large.


Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5811-5825
Author(s):  
Xinhong Zhang

In this paper we study the global dynamics of stochastic predator-prey models with non constant mortality rate and Holling type II response. Concretely, we establish sufficient conditions for the extinction and persistence in the mean of autonomous stochastic model and obtain a critical value between them. Then by constructing appropriate Lyapunov functions, we prove that there is a nontrivial positive periodic solution to the non-autonomous stochastic model. Finally, numerical examples are introduced to illustrate the results developed.


2006 ◽  
Vol 167 (2) ◽  
pp. 246
Author(s):  
Bergström ◽  
Englund ◽  
Leonardsson

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Shukria Akbar ◽  
D. Cole Stevens

AbstractPredation contributes to the structure and diversity of microbial communities. Predatory myxobacteria are ubiquitous to a variety of microbial habitats and capably consume a broad diversity of microbial prey. Predator–prey experiments utilizing myxobacteria have provided details into predatory mechanisms and features that facilitate consumption of prey. However, prey resistance to myxobacterial predation remains underexplored, and prey resistances have been observed exclusively from predator–prey experiments that included the model myxobacterium Myxococcus xanthus. Utilizing a predator–prey pairing that instead included the myxobacterium, Cystobacter ferrugineus, with Pseudomonas putida as prey, we observed surviving phenotypes capable of eluding predation. Comparative transcriptomics between P. putida unexposed to C. ferrugineus and the survivor phenotype suggested that increased expression of efflux pumps, genes associated with mucoid conversion, and various membrane features contribute to predator avoidance. Unique features observed from the survivor phenotype when compared to the parent P. putida include small colony variation, efflux-mediated antibiotic resistance, phenazine-1-carboxylic acid production, and increased mucoid conversion. These results demonstrate the utility of myxobacterial predator–prey models and provide insight into prey resistances in response to predatory stress that might contribute to the phenotypic diversity and structure of bacterial communities.


Parasitology ◽  
2010 ◽  
Vol 137 (6) ◽  
pp. 1027-1038 ◽  
Author(s):  
ANDY FENTON ◽  
SARAH E. PERKINS

SUMMARYPredator-prey models are often applied to the interactions between host immunity and parasite growth. A key component of these models is the immune system's functional response, the relationship between immune activity and parasite load. Typically, models assume a simple, linear functional response. However, based on the mechanistic interactions between parasites and immunity we argue that alternative forms are more likely, resulting in very different predictions, ranging from parasite exclusion to chronic infection. By extending this framework to consider multiple infections we show that combinations of parasites eliciting different functional responses greatly affect community stability. Indeed, some parasites may stabilize other species that would be unstable if infecting alone. Therefore hosts' immune systems may have adapted to tolerate certain parasites, rather than clear them and risk erratic parasite dynamics. We urge for more detailed empirical information relating immune activity to parasite load to enable better predictions of the dynamic consequences of immune-mediated interspecific interactions within parasite communities.


2002 ◽  
Vol 35 (1) ◽  
pp. 107-112 ◽  
Author(s):  
Magno E.M. Meza ◽  
Michel I.S. Costa ◽  
Amit Bhaya ◽  
Eugenius Kaszkurewicz

2011 ◽  
Vol 373 (2) ◽  
pp. 512-520 ◽  
Author(s):  
José Luis Bravo ◽  
Manuel Fernández ◽  
Manuel Gámez ◽  
Bertha Granados ◽  
Antonio Tineo

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