Table

Consumer-resource models. Examples of functions that can be used in the consumer-resource model (3.16), where n1 refers to the level of resources (e.g., number of prey) and n2 refers to the level of consumers (e.g., number of predators).

Function

Description

ftnj = 0

Inflow of resources at a constant rate

finô =~4>

Outflow of resources at a constant rate

finj = r

Constant per capita growth of resource species

f("i) = rn,(l -

Per capita growth of resource species declines linearly with resource level (logistic)

f{n0 = r e~a"'

Per capita growth of resource species declines exponentially with resource level

ginu n2) = a c nxn2

Linear (type I) rate of resource consumption

acni g(n^ = b + n^

Saturating (type II) rate of resource consumption

acn\ Sinx,n2) = kn2 b + n\

Generalized (type III) rate of resource consumption

hin2) = 8 n2

Constant per capita death rate of consumer

hin2) = (8 n2) nz

Per capita death rate of consumer increases linearly with consumer population size

consumers die off at a constant per capita rate 8, so that h(n2) = 8 n2. But again, the rate of change of consumers might involve immigration, emigration, and even births if the consumer can utilize alternative resources. Furthermore, this rate might or might not depend on the current density of consumers. Table 3.3 lists two possible alternatives for h(n2).

By combining the functions f(n,), g(n:,n2), and h(n2) in different ways, a large number of consumer-resource models are possible. Next, we present equations for two of these models, which we will return to in later chapters.

We start with the model that incorporates the simplest choices for each of these functions: a constant inflow (immigration) of resources at rate /"(«,) = 0, a type-I rate of resource consumption£(ni,n2) = u en,n2, and a constant death rate of consumers h(n2) = 8 n2. Equations (3.16) then become

Equation (3.17) can be used, for example, to model the inflow into a lake of a nutrient (e.g., nitrogen) that is required for and limits the growth of an algal species, where ny{t) and n2{t) represent the levels of nutrients and algae, respectively. By setting 6 = 0, equation (3.17) can be used to model the growth of a community when resources begin at some level n,(0) and are not replenished. Interestingly, as we shall see in Chapters 4 and 9, the logistic equation and equation (3.17) without immigration and death (6 = 8 = 0) generate identical predictions about the level of consumers over time.

The second consumer-resource model that we consider is a classic one in ecology, known as the Lotka-Volterra predator-prey model. The only difference from the previous model (3.17) is that the resources are prey that undergo exponential growth in the absence of predators, fin^) = r nt. Equations (3.16) then become rn^t) - acn^t) nz(t),

As we shall see in Chapter 4, the Lotka-Volterra predator-prey model predicts that the number of prey and number of predators cycle over time, demonstrating that species interactions can lead to interesting dynamical behavior that might help explain the cyclic dynamics of several species (e.g., Fussmann et al. 2000; Krebs et al. 1995).

Although we have focused on continuous-time consumer-resource and predator-prey models, discrete-time models are straightforward to develop given the tools that you have now learned. As an example, we develop the discrete-time version of the Lotka-Volterra predator-prey model in Supplementary Material 3.2.

3.5 Epidemiological Models of Disease Spread

The final model that we derive describes the spread of a disease within a population. This is an extension of the flu model that we explored in the previous chapter, but we now add more realism by accounting for births, deaths, recovery, and the gain and loss of immunity. In the previous chapter we considered only two types of individuals, those susceptible to the flu and those infected with the flu. We now allow for a third class of individuals: those who have recovered from the disease and are currently resistant. This model is known as the SIR model in epidemiology, which stands for Susceptible-Infected-Recovered. Consequently, we use the variables S(t), l(t), and R(t) to denote these three types of individuals at time t. We focus on the continuous-time model; the discrete-time version is similar, but with the variables updated after each event.

We begin by using the "mass-action" assumption for the rate at which susceptible individuals contract the disease as in Chapter 2: a c S(t) /(f), where an infected individual contacts susceptible individuals at a rate c per susceptible individual, and a is the probability of disease transmission upon contact (see Figure 2.4c). As illustrated in the flow diagram (Figure 3.9), infected individuals can now recover from the disease at a per capita rate p. We assume that, while infected, these individuals developed antibodies to the pathogen that enable them to resist reinfection. Resistance need not be permanent, however, and individuals that have recovered from the disease become susceptible again at a per capita rate a.

We also incorporate the possibility that individuals die and that infected individuals have a different death rate 8 than healthy individuals, d. Finally, we assume that new susceptible individuals enter the population at a rate 6 through immigration. This assumption is reasonable if immigrants arrive from a location that has not yet been exposed to the disease. The model can also be generalized to account for reproduction from within the population, by replacing 6 with the desired growth rate (e.g., by b(S(t) + /(f) + R(f)) assuming that infection does not alter the birth rate of individuals and that all offspring are born susceptible).

SIR epidemiological models track the dynamics of infectious diseases by modeling the number of susceptible, infected, and recovered individuals within a population.

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