Differences in the nonlinearity of numerical responses

Most saturating functional responses are capable of generating cycles in most mod­els with purely biotic resource growth. This is not true for a saturating numerical response in which consumer growth rate is solely a function of resource abundance.

The parameter z plays the same role in determining the per capita growth rate func­tion for consumer 1 as does the ‘handling time' in a type II functional response (h in eqs (9.1)). However, here z reflects limitations on the growth and/or birth rate of the species rather than limitation on time available for searching for resources. A positive value of z cannot bring about coexistence of the two consumers in the absence of some other factor that generates sustained cycles. In eqs (9.2), there are three potentially time-dependent parameters, which will be examined below. The variation in each time-dependent parameter is assumed not to affect the mean value of the parameter. The simulations described below impose variation using the same multiplicative sinusoidal function with a mean value of 1 given explicitly in eqs (9.1). If all of the functions of time in eqs (9.2) were constants, the system could only have a stable equilibrium with a single consumer present (ignoring the theoretical possibili­ty of a neutral line of equilibria).

The first case considered here is variation in the resource per capita growth rate. Here the parameter r changes sinusoidally and this variation in resource growth leads to sustained cycles in the abundances of all three species. The convex shape of con­sumer 1’s growth function makes it more likely that any type of variation will increase both the mean resource abundance required for the consumer to increase when rare, as well as the long-term mean resource abundance when the nonlinear consumer is present alone. Variation in either or both consumer death rates also cause sus­tained density fluctuations; their effects on the interaction between consumers will be considered later, in Section 9.5.

The parameters used here are equivalent to those used in the main example of eqs (9.1) discussed in Section 9.3. These are: B₁ = B₂ = C₁ = C₂ = 1; z = 10; r = k = 1; d₁ = 0.06. The amplitude of the cycle in r is assumed to have its maximum value, implying sinusoidal variation with a minimum r(t) = 0 and a maximum r(t) = 2. In the system where consumer 1 has a nonlinear functional response (eqs (9.1)) and no variation in r, these parameter values imply large amplitude consumer-resource cycles and a coexistence bandwidth (range of d₂) of approximately 0.357. This com­pares with the maximum range of 1.0 for d₂ that allows existence of the linear species in the absence of competition. On the other hand, a difference in the nonlinearity of the two numerical responses never allows coexistence; the only possibilities in this example are exclusion of consumer 2 for d₂ > 0.15 and exclusion of consumer 1 for d₂ < 0.15.

Table 9.2 describes the outcomes of competition in a system that is identical to that of Table 9.1 except that it is the numerical rather than the functional response of consumer 1 that is nonlinear.

Table 9.2 Resource growth period length and consumer competition outcome; nonlinear numerical response model (eqs. 9.2)

The parameters sharedwith the Table 9.1 model havevalues identical to those in Table 9.1;the one newparameter, z, has a value of 10.

allow coexistence for some range of relative mortality rates in this system without resource growth variation. However, coexistence would not occur (or the self-effects would have to be larger for it to occur) in cases with alternative exclusion outcomes. Thus, one can describe the effects of temporal variation as making coexistence more difficult. (As discussed in Chapter 10, this conclusion maybe reversed if the species occur in a spatially structured metapopulation.)

The nonlinear numerical response system with q = 20 has four categories of out­comes, including a relatively narrow range of d₂ where coexistence of both species or exclusion of the linear species are alternatives.

In summary of the results in Table 9.2, models with nonlinear numerical responses can exhibit coexistence on a single resource in environments with temporally variable resource growth. However, not all types of environmental variation in resource growth allow some coexistence, and variation with a relatively short environmen­tal period can make coexistence more difficult or impossible. Although this is just one example, analysis of several other systems having large differences in numeri­cal response shape also exhibited a wide range of potential effects of resource-driven variation.

9.5