Infinite ODE systems modeling size-structured metapopulations, macroparasitic diseases, and prion proliferation

Infinite systems of ordinary differential equations can describe • spatially implicit metapopulation models with discrete patch-size structure • host-macroparasite models which distinguish hosts by their parasite loads ∗partially supported by NSF grants DMS-0137687 and DMS-0406119 †partially supported by NSF grants DMS-9706787 and DMS-0314529 1 2 Maia Martcheva and Horst R. Thieme • prion proliferation models which distinguish protease-resistant protein aggregates by the number of prion units they contain. It is the aim of this chapter to develop a theory for infinite ODE systems in sufficient generality (based on operator semigroups) and, besides well-posedness, to establish conditions for the solution semiflow to be dissipative and have a compact attractor for bounded sets. For metapopulations, we present conditions for uniform persistence on the one hand and prove on the other hand that a metapopulation dies out, if nobody emigrates from its birth patch or if empty patches are not colonized. Table of contents Introduction 1. The homogeneous linear system: Kolmogorov’s differential equation 2. Solution to the semilinear system 3. General metapopulation models and boundedness of solutions 4. Extinction without migration or colonization of empty patches 5. A more specific metapopulation model 6. Compact attractors 7. Towards the stability of equilibria 8. Instability of every other equilibrium: general result 9. Existence of equilibria and instability of every other equilibrium 10. Stability of the extinction equilibrium versus metapopulation persistence 11. Application to special metapopulation models 12. Special host-macroparasite models and existence of solutions 13. Application to prion proliferation Infinite ODE Systems 3 A. Non-differentiability of the simple birth process semigroup Acknowledgement Bibliography Introduction Infinite systems of ordinary differential equations, w′ =f(t, w, x), xj = ∞ ∑ j=0 αjkxk + gj(t, w, x), j = 0, 1, 2, . . . . (1) where x(t) is the sequence of functions (xj(t)) ∞ j=0, can describe • spatially implicit metapopulation models with discrete patch-size structure [2, 5, 7, 39,43], • host-macroparasite models which distinguish hosts by their parasite loads [6, 13,24,25,34,35,49,50], • prion proliferation models which distinguish protease-resistant protein aggregates by the number of prion units they contain [42,46]. Spatially implicit metapopulation models A metapopulation is a group of populations of the same species which occupy separate areas (patches) and are connected by dispersal. Each separate population in the metapopulation is referred to as a local population. Metapopulations occur naturally or by human activity as a result of habitat loss and fragmentation. In system (1), xj denotes the number of patches with j occupants and w the average number of migrating individuals, or wanderers. The coefficients αjk describe the transition from patches with k occupants to patches with j occupants due to deaths, births and emigration of occupants. The function f gives the rate of change of the number of dispersers due to patch emigration, immigration and disperser death. The functions gj describe the 4 Maia Martcheva and Horst R. Thieme rate of change of the numbers of patches with j occupants due to the immigration of dispersers. The coefficients αjk have the properties typical for infinite transition matrices in stochastic processes with continuous time and discrete state (continuous-time birth and death chains, e.g., see [1] and the references therein). Since they form an unbounded set, existence and uniqueness of solutions to (1) is non-trivial. It is the aim of this chapter to develop a this-related theory in sufficient generality and also establish conditions for the solution semiflow to be dissipative [26], have a compact attractor for bounded sets [26, 53], and be uniformly persistent [5, 27, 57, 59]. We also prove that a metapopulation dies out, if nobody emigrates from its birth patch or if empty patches are not colonized. It is worth mentioning that, though the linear special case xj = ∑∞ k=0 αjkxk can be interpreted as a stochastic model for a population that is not distributed over patches [40], the model (1) is a deterministic model. It inherits the property though that subpopulations on individual patches can become extinct at finite time which is an important feature of real metapopulations. As a trade-off, the metapopulation model (1) is spatially implicit and not able to take spatial heterogeneities into account. A spatially explicit metapopulation model would be a finite system of ordinary differential equations y′ j = ∑N j=1 djkyk +fj(t, y), j = 1, . . . , N , where N is the number of patches and yj the size of the local population on patch j. The coefficients djk would describe the movement from patch k to patch j and the nonlinearities fj the local demographics on patch j due to births and deaths. An example of a spatially explicit metapopulation model (underlying an epidemic model) can be found in Chapter 4. Spatially explicit models can take account of how the patches are situated relatively to each other and of differences between the patches, but do not have the property that a local population can become extinct in finite time. The most basic spatially implicit metapopulation model is the Levins model [37,38] which only considers empty and occupied patches. Incorporating a structure which distinguishes between patches according to local population size makes it possible, e.g., to compare emigration strategies which are based on how crowded a patch is [39]. Alternatively, spatially implicit metapopulation models can be structured by a continuous rather than a discrete variable. This leads to nonlocal partial differential equations or integral equations [23]. The partial differential equations one obtains are similar to those considered in Chapter 1, but have nonlinear terms in the derivative with respect to the size-structure variable. For general information on mathematical metapopulation theory we refer to Infinite ODE Systems 5