| Citation: |
Gamaliel Blé, Iván Loreto–Hernández. TWO-DIMENSIONAL ATTRACTING TORUS IN AN INTRAGUILD PREDATION MODEL WITH GENERAL FUNCTIONAL RESPONSES AND LOGISTIC GROWTH RATE FOR THE PREY[J]. Journal of Applied Analysis & Computation, 2021, 11(3): 1557-1576. doi: 10.11948/20200282
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TWO-DIMENSIONAL ATTRACTING TORUS IN AN INTRAGUILD PREDATION MODEL WITH GENERAL FUNCTIONAL RESPONSES AND LOGISTIC GROWTH RATE FOR THE PREY
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Abstract
The population coexistence in an intraguild food web model is analyzed. Three populations, the prey, the predator and super predator, are considered, where these last two populations are specialists. The sufficient conditions to guarantee a coexistence point, where the intraguild predation model exhibits a zero-Hopf bifurcation, are given. For a wide family of functional responses, these conditions are valid. The numerical simulations varying the functional responses are given. Different limit sets such as, limit cycles or invariant torus are shown.
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Keywords:
- Zero-Hopf bifurcation /
- attracting torus /
- intraguild predation model
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References
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Gamaliel Blé, Iván Loreto–Hernández. TWO-DIMENSIONAL ATTRACTING TORUS IN AN INTRAGUILD PREDATION MODEL WITH GENERAL FUNCTIONAL RESPONSES AND LOGISTIC GROWTH RATE FOR THE PREY[J]. Journal of Applied Analysis & Computation, 2021, 11(3): 1557-1576. doi: 10.11948/20200282
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Figure 1. Attracting torus and graphs of population densities, taking
$ x_0 = 6, \ \ y_0 = 1, \ \ a_1 = 6, \ \ a_3 = 4, \ \ c_3 = c_{30}+\frac{1}{10^9} \ \ { \text{and}} \ \ k = k_0+\frac{4}{10^6}. $ -
Figure 2. Attracting torus, taking
$ x_0 = 3, \ \ y_0 = 1, \ \ a_1 = 3, \ \ a_3 = 4, \ \ c_3 = c_{30}+\frac{1}{10^6} \ \ { \text{and}} \ \ k = k_0-\frac{7}{10^4}. $ - Figure 3. Time series of population densities
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Figure 4. Attracting torus and graphs of population densities, taking
$ x_0 = 10, \ \ y_0 = 1, \ \ a_1 = 10, \ \ a_3 = 4, \ \ c_3 = c_{30}+\frac{1}{10^6} \ \ { \text{and}} \ \ k = k_0-\frac{1}{10^8}. $ - Figure 5. Stable limit cycle and time series
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Figure 6. Attracting torus and graphs of population densities, taking
$ x_0 = 1, \ \ y_0 = \frac{1}{4}, \ \ a_1 = 1, \ \ a_3 = 1, \ \ c_3 = c_{30}-\frac{1}{10^6} \ \ { \text{and}} \ \ k = k_0+\frac{3}{10^3}. $ -
Figure 7. Attracting torus and graphs of population densities, taking
$ x_0 = 1, \ \ y_0 = \frac{1}{4}, \ \ a_1 = 1, \ \ a_3 = 5, \ \ c_3 = c_{30}+\frac{1}{10^6} \ \ { \text{and}} \ \ k = k_0+\frac{1}{10^2}. $ -
Figure 8. Attracting torus and graphs of population densities, taking
$ x_0 = 1, \ \ y_0 = \frac{1}{4}, \ \ a_1 = 1, \ \ a_3 = 5, \ \ c_3 = c_{30}-\frac{1}{10^7} \ \ { \text{and}} \ \ k = k_0+\frac{3}{10^4}. $