Stability and persistence in ODE models for populations with many stages

Jianhong Wu, Horst Thieme, Yijun Lou, Guihong Fan
2015 Mathematical Biosciences and Engineering  
A model of ordinary differential equations is formulated for populations which are structured by many stages. The model is motivated by ticks which are vectors of infectious diseases, but is general enough to apply to many other species. Our analysis identifies a basic reproduction number that acts as a threshold between population extinction and persistence. We establish conditions for the existence and uniqueness of nonzero equilibria and show that their local stability cannot be expected in
more » ... eneral. Boundedness of solutions remains an open problem though we give some sufficient conditions. 2010 Mathematics Subject Classification. Primary: 92D25; Secondary: 34D20, 34D23, 37B25, 93D30. , 41], e.g., and the literature cited there). Stage-structured models are a very natural choice when the life cycle of a species is divided into well distinguishable discrete stages each of which may have their specific climatic and nutritional requirements and their specific vulnerabilities to predators or human control measures. Such models typically lead to systems of ordinary and/or delay differential equations and of difference equations [7, 16, 17 ]. An example par excellence is the tick Ixodes Scpularis that has four main stages: eggs, larvae, nymphs, and adults [27, 28] . All stages except the egg stage have questing, feeding and engorged substages (or phases) with the adults having an additional egg-laying phase and larvae having an additional hardening phase (a total of twelve stages). Adult ticks mainly feed on deer, while nymphs and larvae mainly feed on rodents, and only feeding ticks are able to contract and transmit the infectious diseases like Lyme disease [18, 28] . Questing activity is temperature dependant with adults being active at quite cooler temperatures than larvae and nymphs [26, Fig.3 ]. Only a stage-structured model can hope to catch the impact of these abiotic and biotic factors on the dynamics of a tick population. Densitydependent negative feedback is also stage-specific. Feeding ticks induce an immune reaction of their hosts that increases their mortality, slows down their development, and decreases their fertility with the latter effect being postponed to the egg-laying phase [26, 28] . The model in this manuscript is mainly motivated by tick dynamics, notably by the computer model in [26], but will be formulated general enough to apply to a wide range of stage structured populations. Similarly to [21, 43, 44] , it is a model of many ordinary differential equations (Section 2); for a model of delay-differential equations focussing on ticks see [13] . Our model also applies to epidemic models with many disease stages [20, 30] provided that the equations for susceptible and/or vaccinated individuals can be eliminated. The model incorporates density-dependent feedbacks between the stages that affect mortality, stage-transition, and procreation rates. Our analysis, after establishing uniqueness and global existence of solutions (Section 3), identifies reproduction numbers in a biologically meaningful way and establishes the basic reproduction number as a threshold deciding about extinction or persistence of the population (Sections 4 and 5). We discuss the boundedness of solutions (Section 7) which is a difficult problem if density-dependent negative feedback is exclusively interstage. For this reason, existence of nonzero equilibria is not derived as a consequence of permanence ([23], [34, Ch.6], [45] ), but via fixed point theorems in conical shells [8] (Section 6). Since the systems are large, uniqueness and stability of nonzero equilibria become a challenge (Sections 6.2 and 8). We give an example where a nonzero equilibrium is unstable while the negative feedback is of a very simple nature (Section 8). If a system has several feedbacks, for instance both on stage transition and procreation, then even models with only two stages can show multiple nonzero equilibria and a plethora of complicated bifurcations [2]. We take the difficulties of proving boundedness of solutions as an indication that a literal translation of the computer model in [26] into ordinary differential equations may not capture the negative feedback from adult feeding to adult egg-laying via host immunity or resistance in the right way. We therefore suggest an alternative model formulation in the epilog (Section 9).
doi:10.3934/mbe.2015.12.661 pmid:25974341 fatcat:aub7hvtwmvfzhkse2tebyi6dn4