scholarly journals Transition fronts in nonlocal Fisher-KPP equations in time heterogeneous media

2016 ◽  
Vol 15 (4) ◽  
pp. 1193-1213 ◽  
Author(s):  
Wenxian Shen ◽  
Zhongwei Shen
Keyword(s):  
Heterogeneous Media ◽  
Kpp Equations ◽  
Transition Fronts

scholarly journals Long-time behavior of random and nonautonomous Fisher-KPP equations. Part II. Transition fronts

2019 ◽  
Vol 19 (06) ◽  
pp. 1950046 ◽  
Author(s):  
Rachidi B. Salako ◽  
Wenxian Shen
Keyword(s):  
Time Behavior ◽  
Long Time Behavior ◽  
Spreading Speeds ◽  
Kpp Equation ◽  
Random Transition ◽  
Existence And Stability ◽  
Long Time ◽  
Kpp Equations ◽  
The Stability ◽  
Transition Fronts

In the current series of two papers, we study the long-time behavior of the following random Fisher-KPP equation: [Formula: see text] where [Formula: see text], [Formula: see text] is a given probability space, [Formula: see text] is an ergodic metric dynamical system on [Formula: see text], and [Formula: see text] for every [Formula: see text]. We also study the long-time behavior of the following nonautonomous Fisher-KPP equation: [Formula: see text] where [Formula: see text] is a positive locally Hölder continuous function. In the first part of the series, we studied the stability of positive equilibria and the spreading speeds of (1.1) and (1.2). In this second part of the series, we investigate the existence and stability of transition fronts of (1.1) and (1.2). We first study the transition fronts of (1.1). Under some proper assumption on [Formula: see text], we show the existence of random transition fronts of (1.1) with least mean speed greater than or equal to some constant [Formula: see text] and the nonexistence of random transition fronts of (1.1) with least mean speed less than [Formula: see text]. We prove the stability of random transition fronts of (1.1) with least mean speed greater than [Formula: see text]. Note that it is proved in the first part that [Formula: see text] is the infimum of the spreading speeds of (1.1). We next study the existence and stability of transition fronts of (1.2). It is not assumed that [Formula: see text] and [Formula: see text] are bounded above and below by some positive constants. Many existing results in literature on transition fronts of Fisher-KPP equations have been extended to the general cases considered in the current paper. The current paper also obtains several new results.

scholarly journals Existence, uniqueness and stability of transition fronts of non-local equations in time heterogeneous bistable media

2019 ◽  
Vol 31 (4) ◽  
pp. 601-645
Author(s):  
WENXIAN SHEN ◽  
ZHONGWEI SHEN
Keyword(s):  
Heterogeneous Media ◽  
Periodic Media ◽  
Qualitative Properties ◽  
Transition Front ◽  
Non Local ◽  
The Stability ◽  
Non Local Equations ◽  
Uniquely Ergodic ◽  
Transition Fronts ◽  
Local Dispersal

The present paper is devoted to the study of the existence, the uniqueness and the stability of transition fronts of non-local dispersal equations in time heterogeneous media of bistable type under the unbalanced condition. We first study space non-increasing transition fronts and prove various important qualitative properties, including uniform steepness, stability, uniform stability and exponential decaying estimates. Then, we show that any transition front, after certain space shift, coincides with a space non-increasing transition front (if it exists), which implies the uniqueness, up-to-space shifts and monotonicity of transition fronts provided that a space non-increasing transition front exists. Moreover, we show that a transition front must be a periodic travelling front in periodic media and asymptotic speeds of transition fronts exist in uniquely ergodic media. Finally, we prove the existence of space non-increasing transition fronts, whose proof does not need the unbalanced condition.

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