scholarly journals Phase tipping: how cyclic ecosystems respond to contemporary climate

2021 ◽  
Vol 477 (2254) ◽  
Author(s):  
Hassan Alkhayuon ◽  
Rebecca C. Tyson ◽  
Sebastian Wieczorek
Keyword(s):  
Allee Effect ◽  
Critical Factor ◽  
Global Dynamics ◽  
Tipping Points ◽  
Time Varying ◽  
Predator Prey ◽  
Critical Transitions ◽  
External Conditions ◽  
Predator Prey Models ◽  
Contemporary Climate

We identify the phase of a cycle as a new critical factor for tipping points (critical transitions) in cyclic systems subject to time-varying external conditions. As an example, we consider how contemporary climate variability induces tipping from a predator–prey cycle to extinction in two paradigmatic predator–prey models with an Allee effect. Our analysis of these examples uncovers a counterintuitive behaviour, which we call phase tipping or P-tipping , where tipping to extinction occurs only from certain phases of the cycle. To explain this behaviour, we combine global dynamics with set theory and introduce the concept of partial basin instability for attracting limit cycles. This concept provides a general framework to analyse and identify easily testable criteria for the occurrence of phase tipping in externally forced systems, and can be extended to more complicated attractors.

The global dynamics of stochastic Holling type II predator-prey models with non constant mortality rate

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5811-5825
Author(s):  
Xinhong Zhang
Keyword(s):  
Mortality Rate ◽  
Stochastic Model ◽  
Sufficient Conditions ◽  
Positive Periodic Solution ◽  
Global Dynamics ◽  
Type Ii ◽  
Predator Prey ◽  
Persistence In The Mean ◽  
The Mean ◽  
Predator Prey Models

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.

ALLEE EFFECT, EMIGRATION AND IMMIGRATION IN A CLASS OF PREDATOR-PREY MODELS

BIOMAT 2007 ◽  
2008 ◽  
Author(s):  
EDUARDO GONZÁLEZ-OLIVARES ◽  
JAIME MENA-LORCA ◽  
HÉCTOR MENESES-ALCAY ◽  
BETSABÉ GONZÁLEZ-YAÑEZ ◽  
JOSÉ D. FLORES
Keyword(s):  
Allee Effect ◽  
Emigration And Immigration ◽  
Predator Prey ◽  
Predator Prey Models

ALLEE EFFECT, EMIGRATION AND IMMIGRATION IN A CLASS OF PREDATOR-PREY MODELS

2008 ◽  
Vol 03 (01n02) ◽  
pp. 195-215 ◽  
Author(s):  
EDUARDO GONZÁLEZ-OLIVARES ◽  
JAIME MENA-LORCA ◽  
HÉCTOR MENESES-ALCAY ◽  
BETSABÉ GONZÁLEZ-YAÑEZ ◽  
JOSÉ D. FLORES
Keyword(s):  
Allee Effect ◽  
Initial Conditions ◽  
Stable Limit Cycle ◽  
Threshold Level ◽  
Prey Population ◽  
Stable Limit ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Stable Attractor ◽  
Predator Prey Models

In this work we analyze a predator-prey model proposed by A. Kent et al. in Ecol. Model.162, 233 (2003), in which two aspect of the model are considered: an effect of emigration or immigration on prey population to constant rate and a prey threshold level for predators. We prove that the system when the immigration effect is introduced in the model has a dynamics that is similar to the Rosenzweig-MacArthur model. Also, when emigration is considered in the model, we show that the behavior of the system is strongly dependent on this phenomenon, this due to the fact that trajectories are highly sensitive to the initial conditions, in similar way as when Allee effect is assumed on prey. Furthermore, we determine constraints in the parameters space for which two stable attractor exist, indicating that the extinction of both population is possible in addition with the coexistence of oscillating of populations size in a unique stable limit cycle. We also show that the consideration of a threshold level of prey population for the predator is not essential in the dynamics of the model.

Exploring cyclic dominance of sockeye salmon with a predator–prey model

2014 ◽  
Vol 71 (7) ◽  
pp. 959-972 ◽  
Author(s):  
Christian Guill ◽  
Eddy Carmack ◽  
Barbara Drossel
Keyword(s):  
Sockeye Salmon ◽  
Fraser River ◽  
Predator Prey ◽  
Marine Life ◽  
Predator Prey Model ◽  
Prey Model ◽  
Prey Interaction ◽  
External Conditions ◽  
Predator Prey Models ◽  
Dynamical Pattern

We explain in an intuitive and detailed way a predator–prey model that generates cyclic dominance in Fraser River sockeye salmon. In contrast with usual predator–prey models, this model includes four distinct prey lines and thus a combination of continuous and discrete dynamics, reflecting the particular freshwater and marine life cycle features of sockeye salmon (Oncorhynchus nerka) populations. The predator–prey interaction causing the oscillations takes place in the rearing lakes, rather than in the ocean. The values of most parameters of this model can be estimated from empirical data that are available for the large salmon-rearing lakes in the Fraser River basin. The mechanism that produces the oscillations in this model is compared with other mechanisms that can generate population oscillations, and we argue that predator–prey dynamics is the most likely mechanism to produce the observed patterns. We explain why the period of the oscillation is exactly 4 years, and we explore how the dynamical pattern is affected by changes in external conditions or by management decisions.

scholarly journals Bifurcation analysis of the predator–prey model with the Allee effect in the predator

2021 ◽  
Vol 84 (1-2) ◽  
Author(s):  
Deeptajyoti Sen ◽  
Saktipada Ghorai ◽  
Malay Banerjee ◽  
Andrew Morozov
Keyword(s):  
Allee Effect ◽  
Mathematical Formulation ◽  
Global Bifurcation ◽  
Predator Population ◽  
Stable Limit ◽  
Bifurcation Structure ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Predator Prey Models

AbstractThe 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.

scholarly journals Predator–Prey Models: A Review of Some Recent Advances

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1783
Author(s):  
Érika Diz-Pita ◽  
M. Victoria Otero-Espinar
Keyword(s):  
Global Stability ◽  
Limit Cycles ◽  
Allee Effect ◽  
State Of The Art ◽  
Predator Prey ◽  
Local And Global Stability ◽  
Interacting Species ◽  
Fear Effect ◽  
Recent Advances ◽  
Predator Prey Models

In recent years, predator–prey systems have increased their applications and have given rise to systems which represent more accurately different biological issues that appear in the context of interacting species. Our aim in this paper is to give a state-of-the-art review of recent predator–prey models which include some interesting characteristics such as Allee effect, fear effect, cannibalism, and immigration. We compare the qualitative results obtained for each of them, particularly regarding the equilibria, local and global stability, and the existence of limit cycles.

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