Spatial dynamics of a Lotka-Volterra model with a shifting habitat

Yueding Yuan; Yang Wang; Xingfu Zou

doi:10.3934/dcdsb.2019076

Article Contents

2019, Volume 24, Issue 10: 5633-5671. Doi: 10.3934/dcdsb.2019076

This issue Previous Article Exponential convergence for the 3D stochastic cubic Ginzburg-Landau equation with degenerate noise Next Article Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations

Spatial dynamics of a Lotka-Volterra model with a shifting habitat

1.
School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, China
2.
School of Mathematics and Computer Sciences, Yichun University, Yichun 336000, Jiangxi, China
3.
Department of Applied Mathematics, Western University, London, Ontario, N6A 5B7, Canada

^* Corresponding author
^* Corresponding author

Received: September 30, 2018

Early access: April 2019

Published: August 2019

Research was partially supported by National Natural Science Foundation of China (No. 11561068) and China Postdoctoral Science Foundation (2016M592442) and NSERC of Canada (No. RGPIN-2016-04665).

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Abstract

In this paper, we study a Lotka-Volterra competition-diffusion model that describes the growth, spread and competition of two species in a shifting habitat. Our results show that (Ⅰ) if the competition between the two species are either mutually strong or mutually weak against each other, the spatial dynamics mainly depend on environment worsening speed c and the spreading speed of each species in the absence of the other in the best possible environment; (Ⅱ) if one species is a strong competitor and the other is a weak competitor, then the interplay of the species' competing strengths and the spreading speeds also has an effect on the spatial dynamics. Particularly, we find that a strong but slower competitor can co-persist with a weak but faster competitor, provided that the environment worsening speed is not too fast.

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Mathematics Subject Classification: Primary: 92D40, 92D25; Secondary: 35K57, 93C10.

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Figure 4. Numerical simulations on (6) with (113)- (116): for $ c = 1.8 $ we have $ c> \hat{c}^*(\infty) $ but $ c <c^*(\infty) $, the two species can still co-persist by spreading to the right with the respective asymptotic speeds $ c^*_1(\infty) = 2 $ and $ c^*_2(\infty) = 2.2 $

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Figure 6. Numerical simulations on (6) with (113)- (116) with $ a_2 = 0.36 $ replaced by $ a_2 = 3 $ and the environment worsening rate is very small $ (c = 1.8) $ in the sense that $ c<c_1^*(\infty) $, the two species can still be co-persist by spreading to the right with the respective asymptotic speeds $ c^*_1(\infty) = 2 $ and $ c^*_2(\infty) = 2.2 $

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Figure 1. Numerical simulations on (6) with (113)- (116): when the environment worsening rate is too large ($ c = 2.21 >c^*_i(\infty) > \hat{c}^*_i(\infty) $ for $ i = 1,2 $), both species go to extinct in the habitat

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Figure 2. Numerical simulations on (6) with (113)- (116): when the environment worsening rate is neutral in the sense that $ c = 2.05 \in (c^*_1(\infty), c^*_2(\infty)) $, species 1 becomes extinct in the habitat and species 2 persist by spreading to the right with the asymptotic speed $ c^*_2(\infty) = 2.2 $

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Figure 3. Numerical simulations on (6) with (113)- (116): when the environment worsening rate is very small ($ c = 1.65 $) in the sense that $ c < \hat{c}^*(\infty) $, both species co-persist by spreading to the right with the respective asymptotic speeds $ c^*_1(\infty) = 2 $ and $ c^*_2(\infty) = 2.2 $

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Figure 5. Numerical simulations on (6) with (113)- (116) with $ a_1 = 0.19 $ and $ a_2 = 0.36 $ replaced by $ a_1 = 3 $ and $ a_2 = 2 $ respectively. The environment worsening rate is very small (c = 1.8) in the sense that $ c<c_1^*(\infty) $, species 1 becomes extinct in the habitat and species 2 persist by spreading to the right with the asymptotic speed $ c^*_2(\infty) = 2.2 $

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