Global Classical Solutions to a Predator-Prey Model with Nonlinear Indirect Chemotaxis Mechanism

We deal with the following predator-prey model involving nonlinear indirect chemotaxis mechanism { u t = Δ u + ξ ∇ ⋅ ( u ∇ w ) + a 1 u ( 1 − u r 1 − 1 − b 1 v ) , x ∈ Ω , t > 0 , v t = Δ v − χ ∇ ⋅ ( v ∇ w ) + a 2 v ( 1 − v r 2 − 1 + b 2 u ) , x ∈ Ω , t > 0 , w t = Δ w − w + z γ , x ∈ Ω , t >...

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Bibliographic Details
Published in:Acta applicandae mathematicae 2024-04, Vol.190 (1), p.11, Article 11
Main Authors: Wang, Chang-Jian, Ke, Chun-Hai
Format: Article
Language:English
Subjects:
Applications of Mathematics
Boundary conditions
Calculus of Variations and Optimal Control
Optimization
Charged particles
Computational Mathematics and Numerical Analysis
Equilibrium
Mathematics
Mathematics and Statistics
Partial Differential Equations
Predator-prey simulation
Probability Theory and Stochastic Processes
Vector space
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Summary:We deal with the following predator-prey model involving nonlinear indirect chemotaxis mechanism { u t = Δ u + ξ ∇ ⋅ ( u ∇ w ) + a 1 u ( 1 − u r 1 − 1 − b 1 v ) , x ∈ Ω , t > 0 , v t = Δ v − χ ∇ ⋅ ( v ∇ w ) + a 2 v ( 1 − v r 2 − 1 + b 2 u ) , x ∈ Ω , t > 0 , w t = Δ w − w + z γ , x ∈ Ω , t > 0 , 0 = Δ z − z + u α + v β , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a bounded and smooth domain Ω ⊂ R n ( n ≥ 1 ), where the parameters ξ , χ , a 1 , a 2 , b 1 , b 2 , α , β , γ > 0 . It has been shown that if r 1 > 1 , r 2 > 2 and γ ( α + β ) < 2 n , then there exist some suitable initial data such that the system has a global classical solution ( u , v , w , z ) , which is bounded in Ω × ( 0 , ∞ ) . Compared to the previous contributions, in this work, the boundedness criteria are only determined by the power exponents r 1 , r 2 , α , β , γ and spatial dimension n instead of the coefficients of the system and the sizes of initial data.
ISSN:0167-8019
1572-9036
DOI:10.1007/s10440-024-00648-z