Browsing by Author "Lin, Kuang-Hui"

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Coexistence and extinction for two competing species in patchy environments
(2019-01-01)
Cheng, Chang-Yuan
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Lin, Kuang-Hui
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Shih, Chih-Wen
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應用數學系
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Department of Applied Mathematics
A system of two competing species u and v that diffuse over a two-patch environment is investigated. When u-species has smaller birth rate in the first patch and larger birth rate in the second patch than v-species, and the average birth rate for u-species is larger than or equal to v-species, it was shown in a previous publication that two species coexist in a slow diffusion environment, whereas u-species drives v-species into extinction in a fast diffusion environment. In this paper, we analyze global dynamics and bifurcations for the same model with identical order of birth rates, but with opposite order of average birth rates, i.e., the average birth rate of u-species is less than that of v-species. We observe richer dynamics with two scenarios, depending on the relative difference between the variation in the birth rates of v-species on two patches and the variation in the average birth rates of two species. When the variation in average birth rates is relatively large, there is no stability switch for the semitrivial equilibria. On the other hand, such a stability switch takes place when the variation in average birth rates is relatively mild. In both cases, v-species, with larger average birth rate, prevails in a fast diffusion environment, whereas in a slow diffusion environment, the two species can coexist or u-species that has the greatest birth rate among both species and patches will persist and drive v-species to extinction.
GLOBAL DYNAMICS FOR TWO-SPECIES COMPETITION IN PATCHY ENVIRONMENT
(2014-08-01)
Lin, Kuang-Hui
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Lou, Yuan
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Shih, Chih-Wen
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Tsai, Tze-Hung
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應用數學系
An ODE system modeling the competition between two species in a two-patch environment is studied. Both species move between the patches with the same dispersal rate. It is shown that the species with larger birth rates in both patches drives the other species to extinction, regardless of the dispersal rate. The more interesting case is when both species have the same average birth rate but each species has larger birth rate in one patch. It has previously been conjectured by Gourley and Kuang that the species that can concentrate its birth in a single patch wins if the diffusion rate is large enough, and two species will coexist if the diffusion rate is small. We solve these two conjectures by applying the monotone dynamics theory, incorporated with a complete characterization of the positive equilibrium and a thorough analysis on the stability of the semi-trivial equilibria with respect to the dispersal rate. Our result on the winning strategy for sufficiently large dispersal rate might explain the group breeding behavior that is observed in some animals under certain ecological conditions.
Multiple almost periodic solutions in nonautonomous delayed neural networks
(2007-12-01)
Lin, Kuang-Hui
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Shih, Chih-Wen
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應用數學系
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Department of Applied Mathematics
A general methodology that involves geometric configuration of the network structure for studying multistability and multiperiodicity is developed. We consider a general class of nonautonomous neural networks with delays and various activation functions. A geometrical formulation that leads to a decomposition of the phase space into invariant regions is employed. We further derive criteria under which the n-neuron network admits 2(n) exponentially stable sets. In addition, we establish the existence of 2(n) exponentially stable almost periodic solutions for the system, when the connection strengths, time lags, and external bias are almost periodic functions of time, through applying the contraction mapping principle. Finally, three numerical simulations are presented to illustrate our theory.
Multistability and convergence in delayed neural networks
(2007-01-01)
Cheng, Chang-Yuan
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Lin, Kuang-Hui
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Shih, Chih-Wen
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應用數學系
;
Department of Applied Mathematics
We present the existence of 2(n) stable stationary solutions for a general n-dimensional delayed neural networks with several classes of activation functions. The theory is obtained through formulating parameter conditions motivated by a geometrical observation. Positively invariant regions for the flows generated by the system and basins of attraction for these stationary solutions are established. The theory is also extended to the existence of 2(n) limit cycles for the n-dimensional delayed neural networks with time-periodic inputs. It is further confirmed that quasiconvergence is generic for the networks through justifying the strongly order preserving property as the self-feedback time lags are small for the neurons with negative self-connection weights. Our theory on existence of multiple equilibria is then incorporated into this quasiconvergence for the network. Four numerical simulations are presented to illustrate our theory. (c) 2006 Elsevier B.V. All rights reserved.
Multistability for Delayed Neural Networks via Sequential Contracting
(2015-12-01)
Cheng, Chang-Yuan
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Lin, Kuang-Hui
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Shih, Chih-Wen
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Tseng, Jui-Pin
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應用數學系
In this paper, we explore a variety of new multistability scenarios in the general delayed neural network system. Geometric structure embedded in equations is exploited and incorporated into the analysis to elucidate the underlying dynamics. Criteria derived from different geometric configurations lead to disparate numbers of equilibria. A new approach named sequential contracting is applied to conclude the global convergence to multiple equilibrium points of the system. The formulation accommodates both smooth sigmoidal and piecewise-linear activation functions. Several numerical examples illustrate the present analytic theory.