| Citation: |
Chuihong Yang, Xinan Zhang, Jia Li. COEXISTENCE FROM INTERSPECIFIC MATINGS FOR MOSQUITOES WITH STAGE STRUCTURE[J]. Journal of Applied Analysis & Computation, 2022, 12(3): 1043-1061. doi: 10.11948/20220106
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COEXISTENCE FROM INTERSPECIFIC MATINGS FOR MOSQUITOES WITH STAGE STRUCTURE
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Abstract
Stage-structured models, including groups of larvae and adults, for two interacting mosquito species are studied in this paper. When there are no interspecific matings, the model dynamics follows the competitive exclusion principle such that only one of the two species can survive and there is no coexistent positive equilibrium. As interspecific matings take place, the interactive dynamics become complex. One of the two species can still dominate and drive the other species extinct with no existence of positive equilibrium, either species can survive or die out depending on initial sizes of the species where there exists an unstable positive equilibrium, or the two species coexist with a locally asymptotically stable positive equilibrium under certain conditions. Other dynamical features can occur as well. Detailed mathematical analysis and numerical examples are provided. Brief biological discussions are also given.
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Keywords:
- Mosquito population /
- stage structure /
- interspecific matings /
- competitive exclusion /
- coexistence /
- stability
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References
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Chuihong Yang, Xinan Zhang, Jia Li. COEXISTENCE FROM INTERSPECIFIC MATINGS FOR MOSQUITOES WITH STAGE STRUCTURE[J]. Journal of Applied Analysis & Computation, 2022, 12(3): 1043-1061. doi: 10.11948/20220106
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Figure 1. With parameters given in (4.16) and (4.17), boundary equilibrium
$ E_{01}=(16.6923, 15.4083, 0, 0) $ is locally asymptotically stable and$ E_{02}=(0, 0, 29.5185, 34.3648) $ is unstable. The solution curves of system (4.1) are shown in the left figure. When parameters are given in (4.16) and (4.18), boundary equilibrium$ E_{01}= (12.0769, 11.1479, 0, 0) $ is unstable and$ E_{02}=(0, 0, 19.3952, 22.5795) $ is locally asymptotically stable. The solution curves of system (4.1) are shown in the right figure. In either case, there exists no positive equilibrium. -
Figure 2. With parameters given in (4.32), condition (4.12) is satisfied, system (4.1) has a unique positive equilibrium
$ E^* = (J_1^*, A_1^*, J_2^*, A_2^*) = (10.0343, 13.8937, 33.3241, 25.9928) $ . The two boundary equilibria$ E_{01}=(41.3846, 57.3018, 0, 0) $ and$ E_{02}=(0, 0, 44.7034, 52.0428) $ are both locally asymptotically stable and$ E^* $ is unstable. The phase plane diagram of system (4.19) is shown in the upper figure. The solution curves of system (4.1) are shown in the lower figures where solutions either approach$ E_{01} $ as in the left figure or approach$ E_{02} $ as in the right figure, depending on their initial values. -
Figure 3. With parameters given in (4.33), condition (4.13) is satisfied. System (4.1) has a unique positive equilibrium
$ E^* = (J_1^*, A_1^*, J_2^*, A_2^*) = (42.0476, 58.2197, 24.7103, 19.2470) $ . The two boundary equilibria$ E_{01}=(41.3846, 57.3018, 0, 0) $ and$ E_{02}=(0, 0, 70.0117, 81.5061) $ are both unstable and$ E^* $ is locally asymptotically stable. The phase plane diagram of system (4.19) is shown in the left and the solution curves of system (4.1) are shown in the right. -
Figure 4. With parameters given in (5.1), condition (4.13) is satisfied. System (4.1) has a unique positive equilibrium
$ E^* = (J_1^*, A_1^*, J_2^*, A_2^*) = (3.4685, 1.6065, 0.3923, 0.2744) $ . The two boundary equilibria$ E_{01}=(0.4600, 0.2131, 0, 0) $ and$ E_{02}:=(0, 0, 17.6587, 12.3515) $ are both unstable, and$ E^* $ is also unstable. There exists a locally asymptotically stable periodic solution. The phase plan diagram of system (4.19) is shown in the left and the solution curves of system (4.1) are shown in the right.