| Citation: |
Zhennan Hu, Zhong Li, Fengde Chen, Mengxin He. DYNAMICS OF A LESLIE-GOWER PREDATOR-PREY MODEL WITH SMITH GROWTH AND ALLEE EFFECT[J]. Journal of Applied Analysis & Computation, 2026, 16(4): 1884-1922. doi: 10.11948/20250264
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DYNAMICS OF A LESLIE-GOWER PREDATOR-PREY MODEL WITH SMITH GROWTH AND ALLEE EFFECT
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Abstract
In this paper, considering constant environment and changing environment, we study the bifurcation and transient dynamics of a Leslie-Gower predator-prey model incorporating Smith growth and Allee effect. In a constant environment, we show that the origin is an attractor, and the system admits at most two positive equilibria: One is a saddle and the other can be a weak focus of order four. The unique equilibrium is a saddle-node or a cusp of codimension three. As system parameters vary, the system undergoes a sequence of bifurcations, including saddle-node bifurcations, degenerate Bogdanov-Takens bifurcations of codimension three, and a degenerate Hopf bifurcation of codimension four. In a changing environment, by comparing these transient dynamics with the bifurcation structure of the corresponding constant environment system. We show that the system has rich transient dynamics, such as tracking of unstable equilibria or limit cycles, delay or avoid extinction, and slow and fast regime shifts. We show that low Smith growth or Allee effect is conducive to the stable coexistence of species, while the high Smith growth or Allee effect can eventually lead to the extinction of species.
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Keywords:
- Leslie-Gower /
- Allee effect /
- Smith growth /
- bifurcation /
- changing environment
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References
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Zhennan Hu, Zhong Li, Fengde Chen, Mengxin He. DYNAMICS OF A LESLIE-GOWER PREDATOR-PREY MODEL WITH SMITH GROWTH AND ALLEE EFFECT[J]. Journal of Applied Analysis & Computation, 2026, 16(4): 1884-1922. doi: 10.11948/20250264
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Figure 1.
(a) For system (2.3),
$ (0,0) $ $ (0,\frac{m+s}{s}) $ -
Figure 2.
(a)
$ {E_*} $ $ m=\frac{3}{28},\,q=\frac{1}{4},\,s=\frac{1}{2},\,a=\frac{20}{7} $ $ {E_*} $ $ m=\frac{3}{28},\,q=\frac{1}{4},\,s=\frac{1}{30},\,a=\frac{20}{7} $ $ {E_*} $ $ m=\frac{3}{28},\,q=\frac{1}{4},\,s=\frac{1}{16},\,a=\frac{20}{7} $ $ {E_*} $ $ m=\frac{3}{44},\,q=\frac{23}{44},\,s=\frac{23}{176},\,a=\frac{4}{23} $ -
Figure 3.
Bifurcation diagram of cusp type degenerate Bogdanov-Takens bifurcation of codimension 3 for system (2.1) in the (
$ m,q $ -
Figure 4.
Phase portraits of system (2.1) with
$ (a, s,q) = \left( \frac{4}{23}, \frac{553}{4400},0.45\right) $ -
Figure 5.
Phase portraits of system (2.1) with
$ (a, s,q) = \left( \frac{4}{23}, \frac{553}{4400},0.5045 \right) $ -
Figure 6.
(a) Bifurcation diagram of system (2.1) in the
$ (m,y) $ $ (a, q, s)=(0.17391,0.5122,0.12568) $ -
Figure 7.
Phase portraits of system (2.1) with
$ a=0.17391,q=0.5122,s=0.12568 $ -
Figure 8.
(a) Bifurcation diagram in the
$ (a,x) $ $ (m,q,s)=(0.04534,0.435, $ $ 0.06455) $ -
Figure 9.
Phase portraits of system (2.1) with
$ (m,q,s)=(0.04534,0.435,0.06455) $ -
Figure 10.
Dynamics in system (1.4) for different rate of environmental change with
$ n=0.5 $ $ s=0.525 $ $ m=0.225 $ $ q=1.3 $ $ r=1.5 $ $ a=2 $ $ (x(0),y(0),K(0))=(4,2,6),\, (0.8,0.4,5),\, (5,2.5,26) $ $ (18,9,27) $ -
Figure 11.
Dynamics of system (1.4) for different rates of environmental change with
$ n=1 $ $ s=0.176133 $ $ m=0.5 $ $ q=0.18 $ $ r=0.5 $ $ a=0.67211 $ $ (x(0),y(0),K(0))=(1.5,1.5,4.6),\,(1.5,1.5,5) $ $ (x(0),y(0),K(0))=(1.5,1.5,7) $ -
Figure 12.
Dynamics of system (1.4) for different rate of environmental change
$ \mu $ $ \mu=0.00005 $ $ \mu=-0.000005 $ $ n=1 $ $ s=0.0016 $ $ m=0.4 $ $ q=0.0268 $ $ r=0.483 $ $ a=28 $ -
Figure 13.
Dynamics of system (1.4) in
$ K $ $ x $ $ y $ $ \mu $ $ n=0.5 $ $ s=0.525 $ $ m=0.225 $ $ q=1.3 $ $ r=1.5 $ $ a=2 $ $ (x(0),y(0),K(0))=(0.5,0.25,4.9) $ $ (x(0),y(0),K(0))=(8,4,27) $ -
Figure 14.
Dynamics of system (1.4) in
$ K $ $ x $ $ y $ $ \mu $ $ n=1 $ $ s=0.176133 $ $ m=0.5 $ $ q=0.18 $ $ r=0.5 $ $ a=0.67211 $ $ (x(0),y(0),K(0))=(1.8,1.8,4.7) $ $ (x(0),y(0),K(0))=(2.3,2.3,5.6) $ -
Figure 15.
Dynamics of system (1.4) for different rate of environmental change
$ \mu $ $ \mu=0.00005 $ $ \mu=-0.000005 $ $ n=1 $ $ s=0.0016 $ $ m=0.4 $ $ q=0.0268 $ $ r=0.483 $ $ a=28 $ -
Figure 16.
Dynamics of system (1.4) in
$ K $ $ x $ $ y $ $ \mu $ $ n=0.5 $ $ s=0.525 $ $ m=0.225 $ $ q=1.3 $ $ r=1.5 $ $ a=2 $ $ (x(0),y(0),K(0))=(0.5,0.25,4.9) $ $ (x(0),y(0),K(0))=(8,4,27) $ -
Figure 17.
Dynamics of system (1.4) in
$ K $ $ x $ $ y $ $ \mu $ $ n=1 $ $ s=0.176133 $ $ m=0.5 $ $ q=0.18 $ $ r=0.5 $ $ a=0.67211 $ $ (x(0),y(0),K(0))=(1.8,1.8,4.7) $ $ (x(0),y(0),K(0))=(2.3,2.3,5.6) $