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Conditions for Permanence and Ergodicity of Certain Stochastic Predator-Prey Models
Nguyen Huu Du, Hai Dang Nguyen, George Yin
TL;DR
The paper addresses a gap in conditions for stochastic permanence in an ecology model. It develops sufficient, nearly necessary conditions and establishes stationary distributions, ergodicity, and total-variation convergence, while leaving the critical case λ = 0 open.
Problem
The paper aims to close a gap in establishing sufficient and almost necessary conditions for stochastic permanence.
Method
The analysis treats degenerate cases through total-variation arguments and verification of a Hörmander condition.
Results
λ > 0 yields convergence to a stationary distribution in total variation norm, and a stationary distribution with ergodic property is established.
Takeaways & Limitations
The results establish stationary-distribution and ergodicity conclusions under the stated conditions, including the conclusion associated with λ < 0.
Takeaways & Limitations
The asymptotic behavior in the critical case λ = 0 remains open and requires new techniques.
Abstract
from arXiv · showhide
This work derives sufficient conditions for the permanence and ergodicity of a stochastic predator-prey model with Beddington-DeAngelis functional response. The conditions obtained in fact are very close to the necessary conditions. Both non-degenerate and degenerate diffusions are considered. One of the distinctive features of our results is that our results enables characterization of the support of a unique invariant probability measure. It proves the convergence in total variation norm of the transition probability to the invariant measure. Comparisons to existing literature and related matters to other stochastic predator-prey models are also given.
1 Introduction
The paper studies stochastic predator-prey models with Beddington-DeAngelis functional response, targeting conditions for permanence and ergodicity that are close to necessary. It addresses both non-degenerate and degenerate diffusions and characterizes invariant-measure behavior.
- Beddington-DeAngelis functional response models retain qualitative features of ratio-dependent models while avoiding the low densities problem.
- Previous stochastic predator-prey studies considered Holling, Leslie-Gower, ratio-dependent, and Beddington-DeAngelis functional responses.
- Stochastic permanence describes survival of species forever, and prior conditions for extinction or permanence were restrictive and not close to necessary.
- A major goal is to provide sufficient and almost necessary conditions for permanence and ergodicity in the Beddington-DeAngelis model.
- The analysis includes non-degenerate and degenerate cases, with the degenerate case treated using controllability of associated control systems.
- The results establish invariant-measure existence and uniqueness, total-variation convergence, ergodicity, and characterization of the invariant measure’s support.
2 Threshold Between Extinction and Permanence
The paper derives a threshold λ separating predator extinction from persistence in the stochastic model. When λ is positive, the system is permanent and has a unique ergodic invariant probability measure; when λ is negative, the predator becomes extinct.
- λ < 0 implies that the predator eventually becomes extinct, while x(t) converges weakly to the boundary process’s Gamma stationary distribution.
- λ > 0 yields an invariant probability measure for the two-species process and, under the non-degenerate diffusion, a unique invariant probability measure.
- The positive-threshold regime satisfies stochastic permanence and the strong law of large numbers for integrable functions.
3 Degenerate Case
The degenerate case establishes invariant-measure and ergodicity results through recurrence, control-system accessibility, invariant control sets, and Lie-algebra conditions. Under the stated assumptions, the model has a unique invariant probability measure with characterized support and convergence in total variation.
- Invariant measures: For λ > 0, recurrence and control arguments yield an invariant probability measure on R2,◦ under the relevant parameter conditions.Propositions 3.2 and 3.3 and the preceding lemmas establish recurrence-related results for specified parameter regimes.
- Assumptions and verification: Verifying the nondegeneracy assumption generally involves cumbersome calculations, although direct calculations can verify it for specific parameters.The relevant exceptional points are characterized through polynomial determinant equations involving Lie brackets.
- Invariant control sets: The control system has one invariant control set: R2 when 0 < β < α, and a constrained set when β < 0 or β ≥ α.The latter set is described as {(u, v) : v − β ...} in the supplied passage.
- Invariant measures: Assumption 3.1 implies that the invariant probability measure is unique and that its support is the unique invariant control set C.The paper associates π∗ with the invariant measure of the transformed system and identifies its support with C.
- Ergodicity: Under Assumption 3.2, the transition probability converges to π∗ in total variation as t →∞.Assumption 3.2 is a Hörmander-type Lie-algebra condition requiring dimension 2 at every point.
4 Discussion
The discussion shows that the paper’s extinction, permanence, and ergodicity conditions improve on several existing conditions, while leaving the critical case λ = 0 unresolved. It also identifies parameter regions where ergodicity holds beyond earlier assumptions and extensions to other stochastic models.
- The paper compares its extinction and ergodicity results with existing literature, including conditions for the non-degenerate case.
- λ < 0 is equivalent to the paper’s extinction condition, and the authors state that their predator-extinction result is sharper.
- Theorem 4.1 yields a stationary distribution with an ergodic property under the stated parameter assumptions.
- Ergodicity can hold for all b2 > 0 when λ > 0, including cases where the comparison theorem’s assumption fails for sufficiently small b2.
- For the Holling type-II case, the authors state that their extinction, permanence, and ergodicity conditions are weaker than those in.
- The asymptotic behavior at λ = 0 remains open, although the methods may apply to other functional responses, diffusion coefficients, and Markovian-switching models.