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A predator–prey model with inverse trophic relation and time delay

Authors
Journal
Nonlinear Analysis Real World Applications
1468-1218
Publisher
Elsevier
Publication Date
Volume
6
Issue
4
Identifiers
DOI: 10.1016/j.nonrwa.2004.12.002
Keywords
  • Predator–Prey
  • Inverse Trophic Relation
  • Time-Delay Model
  • Global Attractivity
  • Local Stability
  • Bi-Stable Interior Equilibria

Abstract

Abstract This paper proposes the following time-delay model for an inverse trophic relation where prey x ( t ) eaten by mature predators Y ( t ) can consume the immature predators y ( t ) : x ′ ( t ) = x ( t ) [ r 1 - a 11 x ( t ) + α 1 y ( t ) - a 13 Y ( t ) ] , y ′ ( t ) = - [ r 2 + α 2 x ( t ) ] y ( t ) + a 31 x ( t ) Y ( t ) - a 31 x ( t - τ ) Y ( t - τ ) e ∫ t - τ t [ - r 2 - α 2 x ( s ) ] d s , Y ′ ( t ) = - r 3 Y 2 ( t ) + a 31 x ( t - τ ) Y ( t - τ ) e ∫ t - τ t [ - r 2 - α 2 x ( s ) ] d s , where α 1 , α 2 reflect an inverse trophic relation. Sufficient conditions are obtained for the global attractivity and local stability of an interior equilibrium of the model. It is also shown that α 2 maintains the global properties of the interior equilibrium and that α 1 can cause the existence of bi-stable interior equilibria.

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