## More advanced models for population dynamics

In many biological systems generations overlap so that a population of hosts and parasitoids simultaneously consists of eggs, larvae, pupae and adults. In that case, a more appropriate formulation of the models

Figure 4.10. A diagrammatic formulation of the life history of hosts (horizontal) and parasitoids (vertical) with overlapping generations, useful for the continuous time model. Here TE and TL are the development times of host eggs and larvae; the development time of the parasitoid from egg to adult consists of some time as an egg and development time Tj as a juvenile.

Eggs

Eggs

Figure 4.10. A diagrammatic formulation of the life history of hosts (horizontal) and parasitoids (vertical) with overlapping generations, useful for the continuous time model. Here TE and TL are the development times of host eggs and larvae; the development time of the parasitoid from egg to adult consists of some time as an egg and development time Tj as a juvenile.

involves differential, rather than difference, equations and delays to account for development in the different stages (Murdoch et al. 1987, MacDonald 1989, Briggs 1993). There have been literally volumes written about these approaches; in this section I give a flavor of how the models are formulated and analyzed. In Connections, I point towards more of the literature.

Our goal is to capture the dynamics of hosts and parasitoids in continuous time with overlapping generations. Figure 4.10 should be helpful. After a host egg is laid, there is a development time TE, during which the egg may be attacked by an adult parasitoid. Surviving eggs become larvae and then pupae (both of which are not attacked by the parasitoid) with a development time TL, after which they emerge as adults with average lifetime TA. Parasitoids are characterized in a similar way. It is customary to use different notation to capture the various stages of the host life history, so we now introduce the following variables

E(t) = number of host eggs at time t L(t) = number of host larvae at time t A(t) — number of adult hosts at time t P(t) — number of adult parasitoids at time t

We will derive equations for each of these variables. The rate of change of eggs, dE/dt, is the balance between the rate at which eggs are produced (assumed to be proportional to the adult population size, with no density dependent effects) and the rate at which eggs are lost. Eggs are lost in three ways: due to parasitism (assumed to be proportional to both the number of eggs and the number of parasitoids), due to other sources of mortality, not related to the parasitoid, and due to survival through development and movement into the larval class, which we denote by ME(t), for maturation of eggs at time t. Combining these different rates, we write dE|t) = rA(t) — aP(t)E(t) — ^rp(t) - ME(t) (4.19)

The new parameters in this equation, r, a, and have clear interpretations as the per capita rate at which adults lay eggs, the per capita attack rate by parasitoids and the non-parasitoid related mortality rate. Note that I have written the argument of these equations explicitly on both sides of Eq. (4.19). The reason becomes clear when we explicitly write the maturation function. Eggs that mature into larva at time t had to be laid at time t — TE and survived from then until time t. The rate of egg laying at that earlier time was rA(t — TE) and if we assume random search by parasitoids and other sources of mortality, the probability of survival from the earlier time to time t is exp[— J(i—tE) (aP(s) + yUE)ds]. Combining these, we conclude that

The same kind of logic applies to the larval stage, for which the rate of change of larval numbers is a balance between maturation of eggs into the larval stage, maturation of larvae/pupae into the adult stage, and natural mortality. Hence, we obtain

Adult hosts are produced by maturing larvae and lost due to natural mortality, so that dAdtt) = Ml ( t) —( t) (4.23)

and this completes the description of the host population dynamics.

The reasoning is similar for the parasitoids. Adult parasitoids emerge from eggs that were laid at a time TP before the current time and that survive to produce a parasitoid (assumed to occur with probability a and disappear due to natural mortality), so that we have dp(t) = aaE ( t — TP)P ( t — TP)— ^PP ( t) (4.24)

Equations (4.19)-(4.24) constitute the description of a host para-sitoid system with overlapping generations and potentially different developmental periods. They are called differential-difference equations, for the obvious reason that both derivatives and time differences are involved. What can we say about these kinds of equations in general? Three things. First, finding the steady states of these equations is easy. Second, the numerical solution of these equations is harder than the numerical solution of corresponding solutions without delays (although some software packages might do this automatically for you). Third, the analysis of the stability of these kinds of equations is much, much harder than the work we did in Chapter 2 or in this chapter until now. However, these are important tools so that we now consider a simple version of such an equation and in Connections, I point you towards literature with more details.

Our analysis will focus on the logistic equation with a delay and will follow the treatment given by Murray (2002). We consider the single equation dN

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