scholarly journals Dynamics of Gilpin-Ayala competition model with random perturbation

Filomat ◽  
2010 ◽  
Vol 24 (1) ◽  
pp. 101-113 ◽  
Author(s):  
Maja Vasilova ◽  
Miljana Jovanovic
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solution ◽  
Finite Time ◽  
Random Perturbation ◽  
Competition Model ◽  
Competition System ◽  
Subject Classifications ◽  
Mathematics Subject Classifications

In this paper we study the Gilpin-Ayala competition system with random perturbation which is more general and more realistic than the classical Lotka-Volterra competition model. We verify that the positive solution of the system does not explode in a finite time. Furthermore, it is stochastically ultimately bounded and continuous a.s. We also obtain certain results about asymptotic behavior of the stochastic Gilpin-Ayala competition model. 2010 Mathematics Subject Classifications. 60H10, 34K50. .

Dynamics of an autonomous Gilpin–Ayala competition model with random perturbation

2021 ◽  
pp. 2050043
Author(s):  
Jing Fu ◽  
Qixing Han ◽  
Daqing Jiang ◽  
Yanyan Yang
Keyword(s):  
White Noise ◽  
Numerical Simulations ◽  
Positive Solution ◽  
Necessary Conditions ◽  
Random Perturbation ◽  
Sufficient Conditions ◽  
Competition Model ◽  
Interacting Species ◽  
Sufficient And Necessary Conditions ◽  
Global Positive Solution

This paper discusses the dynamics of a Gilpin–Ayala competition model of two interacting species perturbed by white noise. We obtain the existence of a unique global positive solution of the system and the solution is bounded in [Formula: see text]th moment. Then, we establish sufficient and necessary conditions for persistence and the existence of an ergodic stationary distribution of the model. We also establish sufficient conditions for extinction of the model. Moreover, numerical simulations are carried out for further support of present research.

scholarly journals Asymptotic Behavior of a Stochastic Two-Species Competition Model under the Effect of Disease

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Rong Liu ◽  
Guirong Liu
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solution ◽  
Deterministic Model ◽  
Sufficient Conditions ◽  
Competition Model ◽  
Species Competition ◽  
Martingale Inequality ◽  
Interior Equilibrium ◽  
Exponential Martingale ◽  
Global Positive Solution

This paper is concerned with a stochastic two-species competition model under the effect of disease. It is assumed that one of the competing populations is vulnerable to an infections disease. By the comparison theorem of stochastic differential equations, we prove the existence and uniqueness of global positive solution of the model. Then, the asymptotic pathwise behavior of the model is given via the exponential martingale inequality and Borel-Cantelli lemma. Next, we find a new method to prove the boundedness of the pth moment of the global positive solution. Then, sufficient conditions for extinction and persistence in mean are obtained. Furthermore, by constructing a suitable Lyapunov function, we investigate the asymptotic behavior of the stochastic model around the interior equilibrium of the deterministic model. At last, some numerical simulations are introduced to justify the analytical results. The results in this paper extend the previous related results.

scholarly journals Dynamics of Nonautonomous Stochastic Gilpin-Ayala Competition Model with Jumps

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Yanchao Zang ◽  
Junping Li ◽  
Jiangang Liu
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solution ◽  
Sufficient Conditions ◽  
Competition Model ◽  
Lévy Noise ◽  
Model Driven ◽  
Global Positive Solution ◽  
The Mean ◽  
Pathwise Estimation ◽  
Levy Noise

The nonautonomous stochastic Gilpin-Ayala competition model driven by Lévy noise is considered. First, it is shown that this model has a global positive solution. Then, we discuss the asymptotic behavior of the solution including moment and pathwise estimation. Finally, sufficient conditions for extinction, nonpersistence in the mean, and weak persistence of the solution are established.

scholarly journals Existence and Exact Asymptotic Behavior of Positive Solutions for a Fractional Boundary Value Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Habib Mâagli ◽  
Noureddine Mhadhebi ◽  
Noureddine Zeddini
Keyword(s):  
Asymptotic Behavior ◽  
Continuous Function ◽  
Boundary Value Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Exact Asymptotic Behavior ◽  
Nonnegative Continuous Function

We establish the existence and uniqueness of a positive solution for the fractional boundary value problem , with the condition , where , and is a nonnegative continuous function on that may be singular at or .

scholarly journals Common fixed point theorems for occasionally converse commuting mappings in symmetric spaces

1970 ◽  
Vol 7 (1) ◽  
pp. 56-62
Author(s):  
HK Pathak ◽  
RK Verma
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Symmetric Spaces ◽  
The Other ◽  
Implicit Relation ◽  
Subject Classifications ◽  
Keywords And Phrases ◽  
Mathematics Subject Classifications ◽  
Common Fixed Point Theorems ◽  
Commuting Mappings

In this paper, we introduce the notion of occasionally converse commuting (occ) mappings. Every converse commuting mappings ([1]) are (occ) but the converse need not be true (see, Ex.1.1-1.3). By using this concept, we prove two common fixed point results for a quadruple of self-mappings which satisfy an implicit relation. In first result one pair is (owc) [5] and the other is (occ), while in second result both the pairs are (occ). We illustrate our theorems by suitable examples. Since, there may exist mappings which are (occ) but not conversely commuting, the Theorems 1.1[2], 1.2[2] and 1.3[3] fails to handle those mapping pairs which are only (occ) but not conversely commuting (like Ex.1.4). On the other hand, since every conversely commuting mappings are (occ), so our Theorem 3.1 and 3.2 generalizes these theorems and the main results of Pathak and Verma [6]-[7]   Mathematics Subject Classifications: 47H10; 54H25. Keywords and Phrases: commuting mappings; conversely commuting mappings; occasionally converse commuting (occ) mappings; set of commuting mappings; fixed point. DOI: http://dx.doi.org/ 10.3126/kuset.v7i1.5422 KUSET 2011; 7(1): 56-62  

scholarly journals The role of the range of dispersal in a nonlocal Fisher-KPP equation: an asymptotic analysis

2020 ◽  
pp. 2050032
Author(s):  
Julien Brasseur
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Analysis ◽  
Positive Solution ◽  
Nonlocal Problem ◽  
Type Equation ◽  
Independent Interest ◽  
Kpp Equation ◽  
Open Question

In this paper, we study the asymptotic behavior as [Formula: see text] of solutions [Formula: see text] to the nonlocal stationary Fisher-KPP type equation [Formula: see text] where [Formula: see text] and [Formula: see text]. Under rather mild assumptions and using very little technology, we prove that there exists one and only one positive solution [Formula: see text] and that [Formula: see text] as [Formula: see text] where [Formula: see text]. This generalizes the previously known results and answers an open question raised by Berestycki et al. Our method of proof is also of independent interest as it shows how to reduce this nonlocal problem to a local one. The sharpness of our assumptions is also briefly discussed.

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