4.3 Zooplankton

Step 1: Time Budget for a Foraging Predator

Holling (1959) considered a predator that spends its time either searching for prey or handling (capturing, ingesting, digesting) prey. The total time T is partitioned as:

$$T = T_{\text{search}} + T_{\text{handle}}$$

Step 2: Express Encounters and Handling Time

During search time T_search, the predator encounters prey at rate a (attack rate) proportional to prey density P. Each prey item requires handling time t_h. The number of prey consumed N_e is:

$$N_e = a \cdot P \cdot T_{\text{search}}, \qquad T_{\text{handle}} = N_e \cdot t_h$$

Step 3: Solve for the Type II Response

Substituting T_search = T - N_e t_h into the encounter equation and solving for the per-capita ingestion rate G = N_e/T:

$$N_e = a P (T - N_e t_h) \quad \Rightarrow \quad N_e(1 + a P t_h) = a P T$$

$$\boxed{G = \frac{N_e}{T} = \frac{aP}{1 + aPt_h} = \frac{g_{\max} P}{K_P + P}}$$

where g_max = 1/t_h is the maximum ingestion rate (limited by handling time) and K_P = 1/(a t_h) is the half-saturation constant. This is the Michaelis-Menten form.

Step 4: Type III Response from Density-Dependent Attack Rate

The Type III response arises when the attack rate a itself increases with prey density (due to learning, prey switching, or increased search effort). Setting a = a₀P (linear increase with prey density):

$$G = \frac{a_0 P \cdot P}{1 + a_0 P \cdot P \cdot t_h} = \frac{a_0 P^2}{1 + a_0 t_h P^2} = \frac{g_{\max} P^2}{K_P^2 + P^2}$$

where K_P = 1/sqrt(a₀ t_h). The sigmoidal shape creates a prey refuge at low densities, which has a stabilizing effect on predator-prey dynamics.

Step 5: Verify Limiting Behavior

For both types, we check the limits. At low prey density (P much less than K_P), the Type II response is linear (G ~ aP) while Type III is quadratic (G ~ a₀P²). At high prey density (P much greater than K_P), both saturate at g_max:

$$\text{Type II: } \lim_{P \to \infty} G = g_{\max}, \qquad \text{Type III: } \lim_{P \to \infty} G = g_{\max}$$

Step 1: Prey Growth and Predation Loss

Consider phytoplankton (prey P) growing exponentially at intrinsic rate r in the absence of grazing. Zooplankton (predator Z) consume prey at a rate proportional to the product of both populations (mass-action encounter rate aP Z):

$$\frac{dP}{dt} = rP - aPZ$$

Step 2: Predator Growth from Consumption

Zooplankton convert ingested prey into biomass with efficiency b (gross growth efficiency). In the absence of prey, zooplankton decline at mortality rate m:

$$\frac{dZ}{dt} = baPZ - mZ$$

Step 3: Find Equilibrium Points

Setting both derivatives to zero, the nontrivial equilibrium (coexistence) is:

$$P^* = \frac{m}{ba}, \qquad Z^* = \frac{r}{a}$$

Note that the equilibrium prey density depends on predator parameters (m, b, a) and the equilibrium predator density depends on prey growth rate r. This is the "paradox of enrichment" precursor.

Step 4: Prove Oscillatory Solutions (Conserved Quantity)

The classical Lotka-Volterra system has a conserved quantity (constant of motion) H. Dividing the two ODEs and separating variables:

$$\frac{dZ}{dP} = \frac{baPZ - mZ}{rP - aPZ} = \frac{Z(baP - m)}{P(r - aZ)}$$

$$H = -m\ln P + baP - r\ln Z + aZ = \text{const}$$

Since H is conserved, orbits in the P-Z phase plane are closed curves. The populations oscillate with period T = 2pi/sqrt(rm), and the predator cycle lags the prey cycle by a quarter period.

Step 5: Period and Phase Relationship

Linearizing around the equilibrium (P*, Z*) by setting P = P* + p, Z = Z* + z and keeping first-order terms yields:

$$\frac{d^2p}{dt^2} + rm\,p = 0 \quad \Rightarrow \quad \omega = \sqrt{rm}, \quad T = \frac{2\pi}{\sqrt{rm}}$$

The oscillation period depends only on prey growth rate r and predator mortality m, not on the interaction strength a. This is a key prediction of the classical model.