6.1 Predation and Predator-Prey Dynamics: Ecological Interactions and Mathematical Models

Predation and Exploitative Interactions

Definition and Types of Exploitation

Predation is a key ecological interaction where one organism (the predator) consumes another (the prey), resulting in the prey's death. This is a form of exploitation, which encompasses interactions that benefit one species at the expense of another.

• Exploitation: Interaction between two species where the exploiter's fitness increases while the exploited's fitness decreases.

• Predation: Always results in the mortality of the prey.

• Parasitism and Herbivory: Forms of exploitation that do not always result in the death of the host or plant.

Effects of Predation on Prey

• Lethal Effects: Direct mortality of prey, as seen in classic predator-prey interactions (e.g., lions hunting zebras).

• Natural Selection: Predation pressure drives the evolution of prey adaptations to avoid being eaten.

Prey Defenses Against Predation

Physical and Chemical Defenses

• Physical Defenses: Structures such as spines, thorns, shells, and stingers (e.g., porcupines, snails).

• Crypsis: Camouflage or background matching to avoid detection (e.g., stick-like caterpillars).

• Mimicry: Imitation of unpalatable or dangerous species (e.g., caterpillars mimicking snakes).

• Warning Coloration (Aposematism): Bright colors signaling toxicity or danger (e.g., blue-ringed octopus, poison dart frog, coral snake).

• Chemical Defenses: Production of toxins or deterrent compounds (e.g., plant phenolics, animal venoms).

Refuges and Behavioral Defenses

• Size Refuge: Prey outgrow the size range vulnerable to predators (e.g., large daphnia escape mussel predation).

• Spatial Refuge: Prey use complex habitats to avoid predators (e.g., sunfish feeding in vegetation).

• Group Living: Safety in numbers (e.g., schools of fish, meerkat colonies).

• Experimental Evidence: Huffaker's mite experiments showed that spatial complexity increases prey persistence.

Inducible Defenses and Non-Lethal Effects

Prey may exhibit inducible defenses—changes in response to predator presence that do not require direct attack.

• Physical Changes: Altered coloration (e.g., chameleons), increased shell thickness (e.g., snails), more spines (e.g., acacia), increased silica or phenolics in plants.

• Behavioral Changes: Reduced activity (e.g., lizards), freezing behavior (e.g., deer), habitat shifts (e.g., daphnia changing depth, snails leaving water).

• Life History Changes: Altered growth rates (e.g., fish growing faster to escape gape-limited predators), changes in emergence or hatching time (e.g., tadpoles hatching early to escape snakes).

Predator Adaptations and the Evolutionary Arms Race

Predator Adaptations

• Physical Adaptations: Morphological traits such as sharp teeth, speed (e.g., cheetahs), or resistance to toxins (e.g., honey badgers vs. cobras).

• Chemical Adaptations: Ability to detoxify or withstand prey defenses.

• Behavioral Adaptations: Adjusting hunting strategies to prey activity patterns.

Evolutionary Arms Race

Predator-prey interactions are evolutionarily dynamic. As prey evolve better defenses, predators evolve better ways to overcome them, resulting in a continuous 'arms race.'

Mathematical Models: Lotka-Volterra Predator-Prey Equations

Model Overview

The Lotka-Volterra equations describe the population dynamics of predator and prey species. They are foundational in ecology for understanding cycles and stability in biological communities.

Prey Population Equation

• Let N = prey population size, P = predator population size

• r = intrinsic rate of increase of prey

• a = capture efficiency (rate at which predators capture prey)

Equation:

$dN/dt = rN - aNP$

Predator Population Equation

• b = efficiency of converting captured prey into predator offspring

• m = predator mortality rate

Equation:

$dP/dt = baNP - mP$

Parameter Definitions

• r: Intrinsic rate of increase of prey

• a: Capture efficiency

• b: Conversion efficiency (prey to predator offspring)

• m: Predator mortality rate

Assumptions of the Lotka-Volterra Model

• Prey growth is limited only by predation (no other limiting factors).

• Predators are specialists, feeding only on the modeled prey species.

• Predators can consume an infinite number of prey (no satiation).

• Predators and prey encounter each other randomly in a homogeneous environment (no refuges or behavioral avoidance).

Note: These assumptions are rarely met in nature but provide a foundation for more complex models.

Isoclines and Population Dynamics

Zero Growth Isoclines

Isoclines are lines in population space where the growth rate of one population is zero.

• Prey Isocline: Where prey population growth is zero ($dN/dt = 0$)

• Predator Isocline: Where predator population growth is zero ($dP/dt = 0$)

Solving for Isoclines

• Set $dN/dt = 0$ in the prey equation:

$0 = rN - aNP$

Solving for P:

$P = \frac{r}{a}$

• Set $dP/dt = 0$ in the predator equation:

$0 = baNP - mP$

Solving for N:

$N = \frac{m}{ba}$

Interpreting Isoclines

• Below the prey isocline ($P < r/a$), prey population increases; above, it decreases.

• To the right of the predator isocline ($N > m/ba$), predator population increases; to the left, it decreases.

Population Dynamics and Cycles

The interaction of predator and prey isoclines creates four quadrants, each with different population trends. The resulting vectors in each quadrant produce cyclical population dynamics:

• Both populations can increase or decrease depending on their position relative to the isoclines.

• These cycles are characterized by time lags: predator populations lag behind prey populations.

Graphical Representation

Population cycles can be visualized as oscillations in predator and prey densities over time, with predator peaks following prey peaks.

Real-World Example: Lynx and Hare Cycles

• Long-term data on lynx and hare populations in the Arctic show regular cycles with time lags between prey and predator peaks.

• Hare reproductive rates peak before hare densities, and both decline after predator numbers increase.

• These patterns are consistent with Lotka-Volterra predictions.

Worked Example: Wolves and Moose in Alaska

Given:

• Prey (moose) intrinsic rate of increase, $r = 0.1$

• Predator (wolf) mortality rate, $m = 0.3$

• Capture efficiency, $a = 0.002$

• Conversion efficiency, $b = 1.5$

• Initial populations: 200 moose, 150 wolves

Calculate Isoclines

• Predator isocline (number of moose needed to sustain wolf population):

$N = \frac{m}{ba} = \frac{0.3}{1.5 \times 0.002} = 100$

• Prey isocline (number of wolves needed to keep moose in check):

$P = \frac{r}{a} = \frac{0.1}{0.002} = 50$

• Current populations: 200 moose (N), 150 wolves (P)

Interpretation

• Moose population is above the predator isocline (N > 100), so wolves have enough prey to increase.

• Wolf population is above the prey isocline (P > 50), so moose population will decrease.

• However, in the lecture, the calculation for the predator isocline was $N = 225$ (using $m = 0.3$, $b = 1.5$, $a = 0.002$):

$N = \frac{0.3}{1.5 \times 0.002} = \frac{0.3}{0.003} = 100$

Additional info: The lecture used $N = 225$; this may be a rounding or transcription error. The calculation above is based on the provided parameters.

• With 200 moose and 150 wolves, both populations are above their isoclines, so both are decreasing (as per the lecture's quadrant analysis).

Summary Table: Lotka-Volterra Predator-Prey Model Parameters

Parameter	Symbol	Definition	Example Value
Prey population size	N	Number of prey individuals	200 (moose)
Predator population size	P	Number of predator individuals	150 (wolves)
Intrinsic rate of increase (prey)	r	Prey population growth rate without predation	0.1
Capture efficiency	a	Rate at which predators capture prey	0.002
Conversion efficiency	b	Efficiency of converting prey into predator offspring	1.5
Predator mortality rate	m	Rate at which predators die	0.3

Key Takeaways

• Predation is a major ecological force shaping prey adaptations and population dynamics.

• Prey employ a variety of defenses, both physical and behavioral, to avoid predation.

• Predator-prey interactions can be modeled mathematically using the Lotka-Volterra equations, which predict population cycles and time lags.

• Real-world data (e.g., lynx and hare) support the existence of such cycles, though real systems are more complex than the basic model.

Additional info: In practice, factors such as alternative prey, prey refuges, and environmental heterogeneity modify these dynamics, but the Lotka-Volterra model remains a foundational concept in ecology.