##
**Permanence and extinction in logistic and Lotka-Volterra systems with diffusion.**
*(English)*
Zbl 0985.34061

The authors study the effect of diffusion on the permanence and extinction of endangered species that live in changing patch environment. Two models are considered. The first one is a logistic system with diffusion
\[
\dot{x}_{i}=x_{i}[b_{i}(t)-a_{i}(t)x_{i}]+ \sum\limits_{j=1}^{n}D_{ij}(t)(x_{j}-\alpha_{ij}(t)x_{i}), \;\;i=1,\ldots,n,\tag{1}
\]
where \(x_{i}\), \(i=1,\ldots,n\), denotes the species \(x\) in the patch \(i\), \(b_{i}(t)\) belongs to the set \(C\) of all bounded continuous functions on \(\mathbb{R}\), \(a_{i}(t)\), \(D_{ij}(t)\) belong to the set \(C_{+}\subset C\) of continuous functions bounded below by a positive constant, \(D_{ii}(t)\equiv 0\). \(b_{i}(t)\) is the intrinsic growth rate for species \(x\) in patch \(i\), \(a_{i}(t)\) represents the self-inhibition coefficient, \(D_{ij}(t)\) is the diffusion coefficient of species \(x\) from patch \(j\) to patch \(i\). The parameter \(\alpha_{ij}(t)\in C\) corresponds to the boundary conditions of the continuous diffusion case.

The system of differential equations is said to be permanent if there exists a compact set \(K\) in the interior of \({\mathbb{R}^{n}_{+}=\{x\in\mathbb{R}^{n}:x_{i}\geq 0, i=1,\ldots,n\}}\), such that all solutions starting in the interior of \(\mathbb{R}^{n}_{+}\) ultimately enter \(K\). Denote by \[ A_{L}(f)=\lim\limits_{r\to\infty}\inf\limits_{t-s\geq r} \frac{1}{t-s}\int_{s}^{t}f(\tau) d\tau \] the lower average of \({f\in C}\).

The following results concerning system (1) are obtained: There exists \({M>0}\) such that the region \({D=\{0<x_{i}\leq M,i=1,\ldots,n\}}\) is positively invariant with respect to (1). Moreover, any solution to (1) with \({x(0)>0}\) is also ultimately bounded below away from zero provided that one of the following conditions is satisfied.

(H.1) There exists \({i_{0}\in\{1,\ldots,n\}}\) such that \({A_{L}(\theta)>0}\), with \(\theta(t)=b_{i_{0}}(t)-\sum_{j=1}^{n}D_{i_{0}j}(t) \alpha_{i_{0}j}(t)\).

(H.2) \({A_{L}(\phi)>0}\), with \( \phi(t)=\min_{1\leq i\leq n} \{b_{i}-\sum _{j=1}^{n} D_{ij}(t)\alpha_{ij}(t)+ \sum _{j=1}^{n}D_{ji}(t) \}. \)

It means that under these hypotheses system (1) is permanent.

If all coefficients in (1) are periodic with a common period \({\omega}\), then at least one positive \(\omega\)-periodic solution exists. The uniqueness and stability of the periodic solution is also discussed.

In section 4, ehe authors consider the Lotka-Volterra diffusion model \[ \begin{aligned} \dot{x}_{i}&=x_{i}[b_{i}(t)-a_{i}(t)x_{i}-c_{i}(t) y_{i}]+ \sum_{j=1}^{n}D_{ij}(t)(x_{j}- \alpha_{ij}(t)x_{i}),\\ \dot{y}_{i}&=y_{i}[-d_{i}(t)+e_{i}(t)x_{i}- f_{i}(t)y_{i}]+ \sum_{j=1}^{n}\lambda_{ij}(t)(y_{j}- \beta_{ij}(t)y_{i}),\end{aligned} \qquad i=1,\ldots,n, \tag{2} \] where \(y_{i}\) is the density of predator species \(y\) in patch \(i\) and \(d_{i}(t)\), \(e_{i}(t)\), \(c_{i}(t)\) and \(\beta_{ij}(t)\) are all nonnegative and bounded continuous functions. Furthermore, \(f_{i}(t), \lambda_{ij}(t) \in C_{+}\), \(i\neq j\) and \(\lambda_{ii}(t)\equiv 0\).

Under hypotheses analogous to conditions (H.1) and (H.2) given above, the permanence of predator and prey species in model (2) is proved.

The effect of a competitor \(y\) and diffusion on the survival of the native endangered species \(x\) is studied in section 5.

The biological meaning of the main results is discussed in the final section. The authors conclude that the diffusion rates play an important role in the determination of the permanence and extinction of single and multiple endangered species that live in weak patchy environments. Some illustrative examples are given.

The system of differential equations is said to be permanent if there exists a compact set \(K\) in the interior of \({\mathbb{R}^{n}_{+}=\{x\in\mathbb{R}^{n}:x_{i}\geq 0, i=1,\ldots,n\}}\), such that all solutions starting in the interior of \(\mathbb{R}^{n}_{+}\) ultimately enter \(K\). Denote by \[ A_{L}(f)=\lim\limits_{r\to\infty}\inf\limits_{t-s\geq r} \frac{1}{t-s}\int_{s}^{t}f(\tau) d\tau \] the lower average of \({f\in C}\).

The following results concerning system (1) are obtained: There exists \({M>0}\) such that the region \({D=\{0<x_{i}\leq M,i=1,\ldots,n\}}\) is positively invariant with respect to (1). Moreover, any solution to (1) with \({x(0)>0}\) is also ultimately bounded below away from zero provided that one of the following conditions is satisfied.

(H.1) There exists \({i_{0}\in\{1,\ldots,n\}}\) such that \({A_{L}(\theta)>0}\), with \(\theta(t)=b_{i_{0}}(t)-\sum_{j=1}^{n}D_{i_{0}j}(t) \alpha_{i_{0}j}(t)\).

(H.2) \({A_{L}(\phi)>0}\), with \( \phi(t)=\min_{1\leq i\leq n} \{b_{i}-\sum _{j=1}^{n} D_{ij}(t)\alpha_{ij}(t)+ \sum _{j=1}^{n}D_{ji}(t) \}. \)

It means that under these hypotheses system (1) is permanent.

If all coefficients in (1) are periodic with a common period \({\omega}\), then at least one positive \(\omega\)-periodic solution exists. The uniqueness and stability of the periodic solution is also discussed.

In section 4, ehe authors consider the Lotka-Volterra diffusion model \[ \begin{aligned} \dot{x}_{i}&=x_{i}[b_{i}(t)-a_{i}(t)x_{i}-c_{i}(t) y_{i}]+ \sum_{j=1}^{n}D_{ij}(t)(x_{j}- \alpha_{ij}(t)x_{i}),\\ \dot{y}_{i}&=y_{i}[-d_{i}(t)+e_{i}(t)x_{i}- f_{i}(t)y_{i}]+ \sum_{j=1}^{n}\lambda_{ij}(t)(y_{j}- \beta_{ij}(t)y_{i}),\end{aligned} \qquad i=1,\ldots,n, \tag{2} \] where \(y_{i}\) is the density of predator species \(y\) in patch \(i\) and \(d_{i}(t)\), \(e_{i}(t)\), \(c_{i}(t)\) and \(\beta_{ij}(t)\) are all nonnegative and bounded continuous functions. Furthermore, \(f_{i}(t), \lambda_{ij}(t) \in C_{+}\), \(i\neq j\) and \(\lambda_{ii}(t)\equiv 0\).

Under hypotheses analogous to conditions (H.1) and (H.2) given above, the permanence of predator and prey species in model (2) is proved.

The effect of a competitor \(y\) and diffusion on the survival of the native endangered species \(x\) is studied in section 5.

The biological meaning of the main results is discussed in the final section. The authors conclude that the diffusion rates play an important role in the determination of the permanence and extinction of single and multiple endangered species that live in weak patchy environments. Some illustrative examples are given.

Reviewer: Oleg Anashkin (Simferopol)

### MSC:

34K13 | Periodic solutions to functional-differential equations |

92D25 | Population dynamics (general) |

34K05 | General theory of functional-differential equations |

34K20 | Stability theory of functional-differential equations |

34K25 | Asymptotic theory of functional-differential equations |

34K10 | Boundary value problems for functional-differential equations |

### Keywords:

logistic equation; Lotka-Volterra system; diffusion; permanence; extinction; periodic solution; stability
PDF
BibTeX
XML
Cite

\textit{J. Cui} and \textit{L. Chen}, J. Math. Anal. Appl. 258, No. 2, 512--535 (2001; Zbl 0985.34061)

Full Text:
DOI

### References:

[1] | Allen, L.J.S., Persistence and extinction in single-species reaction – diffusion models, Bull. math. biol., 45, 209-227, (1983) · Zbl 0543.92020 |

[2] | Allen, L.J.S., Persistence, extinction, and critical patch number for island populations, J. math. biol., 24, 617-625, (1987) · Zbl 0603.92019 |

[3] | Beretta, E.; Takeuchi, Y., Global stability of single-species diffusion Volterra models with continuous time delays, Bull. math. biol., 49, 431-448, (1987) · Zbl 0627.92021 |

[4] | Beretta, E.; Takeuchi, Y., Global asymptotic stability of lotka – volterra diffusion models with continuous time delays, SIAM J. appl. math., 48, 627-651, (1988) · Zbl 0661.92018 |

[5] | Beretta, E.; Solimano, F., Global stability and periodic orbits for two patch predator – prey diffusion delay models, Math. biosci., 85, 153-183, (1987) · Zbl 0634.92017 |

[6] | Cao, F.; Chen, L., Asymptotic behavior of nonautonomous diffusive lotka – volterra model, System sci. & math. sci., 11, 107-111, (1998) · Zbl 0913.92017 |

[7] | Cushing, J.M., Integro-differential equations and delay models in population dynamics, Lecture notes in biomathematics, 20, (1977), Springer-Verlag Berlin · Zbl 0363.92014 |

[8] | Freedman, H.I.; Waltman, P., Mathematical models of population interaction with dispersal. I. stability of two habitats with and without a predator, SIAM J. math., 32, 631-648, (1977) · Zbl 0362.92006 |

[9] | Freedman, H.I., Single species migration in two habitats: persistence and extinction, Math. model., 8, 778-780, (1987) |

[10] | Freedman, H.I.; Takeuchi, Y., Global stability and predator dynamics in a model of prey dispersal in a patchy environment, Nonlinear anal. theory methods appl., 13, 993-1002, (1989) · Zbl 0685.92018 |

[11] | Hastings, A., Spatial heterogeneity and the stability of predator prey systems, Theoret. popul. biol., 12, 37-48, (1977) · Zbl 0371.92016 |

[12] | Krasnoselskii, M.A., Translation along trajectories of differential equations, Translations of math. monographs, 19, (1968), American Mathematical Society Providence |

[13] | Kuang, Y.; Takeuchi, Y., Predator-prey dynamics in models of prey dispersal in two-patch environments, Math. biosci., 120, 77-98, (1994) · Zbl 0793.92014 |

[14] | Levin, S.A., Dispersion and population interactions, Amer. natur., 108, 207-228, (1974) |

[15] | Levin, S.A.; Segel, L.A., Hypothesis to explain the origin of planktonic patchness, Nature, 259, 659, (1976) |

[16] | Lu, Z.; Takeuchi, Y., Global asymptotic behavior in single-species discrete diffusion systems, J. math. biol., 32, 67-77, (1993) · Zbl 0799.92014 |

[17] | Mahbuba, R.; Chen, L., On the nonautonomous lotka – volterra competition system with diffusion, Differential equations dynam. systems, 2, 243-253, (1994) · Zbl 0874.34048 |

[18] | Massera, J.L., The existence of periodic solution of differential equations, Duke math. J., 17, 457-475, (1950) · Zbl 0038.25002 |

[19] | Skellam, J.D., Random dispersal in theoretical population, Biometrika, 38, 196-216, (1951) · Zbl 0043.14401 |

[20] | Smith, H.L., Cooperative systems of differential equations with concave nonlinearities, Nonlinear anal., 10, 1037-1052, (1986) · Zbl 0612.34035 |

[21] | Takeuchi, Y., Global stability in generalized lotka – volterra diffusion systems, J. math. anal. appl., 116, 209-221, (1986) · Zbl 0595.92013 |

[22] | Takeuchi, Y., Diffusion effect on stability of lotka – volterra model, Bull. math. biol., 46, 585-601, (1986) · Zbl 0613.92025 |

[23] | Takeuchi, Y., Cooperative system theory and global stability of diffusion models, Acta appl. math., 14, 49-57, (1989) · Zbl 0665.92017 |

[24] | Takeuchi, Y., Diffusion-mediated persistence in two-species competition lotka – volterra model, Math. biosci., 95, 65-83, (1989) · Zbl 0671.92022 |

[25] | Takeuchi, Y., Conflict between the need to forage and the need to avoid competition: persistence of two-species model, Math. biosci., 99, 181-194, (1990) · Zbl 0703.92024 |

[26] | Tineo, A., An iterative scheme for the N-competing species problem, J. differential equations, 116, 1-15, (1995) · Zbl 0823.34048 |

[27] | Vance, R.R., The effect of dispersal on population stability in one-species, discrete space population growth models, Amer. natur., 123, 230-254, (1984) |

[28] | Wang, W.; Chen, L., Global stability of a population dispersal in a two-patch environment, Dynamic systems appl., 6, 207-216, (1997) · Zbl 0892.92026 |

[29] | Xun, Y., State, disturbance, and development of Chinese giant pandas, Chinese wildlife, 18, 9-11, (1990) |

[30] | Yange, Y.; Kuanwui, W.; Tiejun, W., Giant Panda’s moving habit in poping, Acta theridogica sinica, 14, 9-14, (1994) |

[31] | Yucun, C.; Aimin, W., The urgent problems in reproduction of giant pandas, Chinese wildlife, 79, 3-5, (1994) |

[32] | Zeng, G.; Chen, L.; Chen, J., Persistence and periodic orbits for two-species nonautonomous diffusion lotka – volterra models, Math. comput. modelling, 20, 69-80, (1994) · Zbl 0827.34040 |

[33] | Zhang, J.; Chen, L., Periodic solutions of single-species nonautonomous diffusion models with continuous time delays, Math. comput. modelling, 23, 17-27, (1996) · Zbl 0864.60058 |

[34] | Zhou, Y., Analysis on decline of wild alligator sinensis population, Sichuan J. zool., 16, 137-139, (1997) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.