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In this paper, an insect-parasite-host model with logistic growth of triatomine bugs is formulated to study the transmission between hosts and vectors of the Chagas disease by using dynamical system approach. We derive the basic reproduction numbers for triatomine bugs and Trypanosoma rangeli as two thresholds. The local and global stability of the vector-free equilibrium, parasite-free equilibrium and parasite-positive equilibrium is investigated through the derived two thresholds. Forward bifurcation, saddle-node bifurcation and Hopf bifurcation are proved analytically and illustrated numerically. We show that the model can lose the stability of the vector-free equilibrium and exhibit a supercritical Hopf bifurcation, indicating the occurrence of a stable limit cycle. We also find it unlikely to have backward bifurcation and Bogdanov-Takens bifurcation of the parasite-positive equilibrium. However, the sustained oscillations of infected vector population suggest that Trypanosoma rangeli will persist in all the populations, posing a significant challenge for the prevention and control of Chagas disease.
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A Collective Colony Migration Model with Hill Functions in Recruitment
Social insect colonies’ robust and efficient collective behaviors without any central control contribute greatly to their ecological success. Colony migration is a leading subject for studying collective decision-making in migration. In this paper, a general colony migration model with Hill functions in recruitment is proposed to investigate the underlying decision making mechanism and the related dynamical behaviors. Our analysis provides the existence and stability of equilibrium, and the global dynamical behavior of the system. To understand how piecewise functions and Hill functions in recruitment impact colony migration dynamics, the comparisons are performed in both analytic results and bifurcation analysis. Our theoretical results show that the dynamics of the migration system with Hill functions in recruitment differs from that of the migration system with piecewise functions in the following three aspects: (1) all population components in our colony migration model with Hill functions in recruitment are persistent; (2) the colony migration model with Hill functions in recruitment has saddle and saddle-node bifurcations, while the migration system with piecewise functions does not; (3) the system with Hill functions has only equilibrium dynamics, i.e. either has a global stability at one interior equilibrium or has bistablity among two locally stable interior equilibria. Bifurcation analysis shows that the geometrical shape of the Hill functions greatly impacts the dynamics: (1) the system with flatter Hill functions is less likely to exhibit bistability; (2) the system with steeper functions is prone to exhibit bistability, and the steady state of total active workers is closer to that of active workers in the system with piecewise function.
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- PAR ID:
- 10383937
- Date Published:
- Journal Name:
- International Journal of Bifurcation and Chaos
- Volume:
- 32
- Issue:
- 14
- ISSN:
- 0218-1274
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1142/S0218127422502133
