12.1 Population dynamics and predator-prey models

Population dynamics and predator-prey models are key concepts in biology. They help us understand how species interact and populations change over time. These models use math to predict population growth, decline, and cycles in nature.

Logistic growth and carrying capacity are central ideas in population dynamics. Predator-prey interactions, described by Lotka-Volterra equations, show how two species affect each other's numbers. These models reveal fascinating patterns in ecosystems and help manage wildlife populations.

Population Growth Models

Logistic Growth and Carrying Capacity

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• Logistic growth models population dynamics with limited resources
• Incorporates density-dependent factors that limit population growth
• Carrying capacity ($K$) represents the maximum population size an environment can sustain
• As population size approaches carrying capacity, growth rate decreases due to resource limitations (food, space)
• Logistic growth equation: $\frac{dN}{dt} = rN(1-\frac{N}{K})$, where $N$ is population size, $r$ is intrinsic growth rate, and $K$ is carrying capacity
• Logistic growth exhibits sigmoid (S-shaped) curve, with slow initial growth, rapid intermediate growth, and slow growth near carrying capacity

Density-Dependent Factors Affecting Population Growth

• Density-dependent factors have a greater impact on population growth as population density increases
• Includes competition for resources, predation, disease transmission, and space limitations
• Resource competition intensifies as population approaches carrying capacity, limiting individual growth and reproduction
• Increased population density facilitates the spread of diseases, potentially reducing population size
• Predation pressure may increase with higher prey population density, regulating population growth
• Limited space can lead to increased stress, reduced reproduction, and emigration from the population

Predator-Prey Interactions

Lotka-Volterra Equations and Predator-Prey Cycles

• Lotka-Volterra equations describe the dynamics of predator-prey interactions
• Consists of two coupled differential equations: one for prey population growth and another for predator population growth
• Prey equation: $\frac{dN}{dt} = rN - aNP$, where $N$ is prey population size, $r$ is prey growth rate, $a$ is predation rate, and $P$ is predator population size
• Predator equation: $\frac{dP}{dt} = baNP - mP$, where $b$ is conversion efficiency of prey to predator, and $m$ is predator mortality rate
• Predator-prey cycles emerge from the interactions between the two populations
• As prey population increases, predator population grows due to increased food availability
• Growing predator population reduces prey population through increased predation
• Declining prey population leads to a decrease in predator population due to reduced food availability
• The cycle repeats, resulting in oscillations in both predator and prey populations (lynx-hare cycles)

Functional Response and Equilibrium Points

• Functional response describes the relationship between prey density and predator consumption rate
• Type I functional response: linear increase in predation rate with increasing prey density until saturation
• Type II functional response: predation rate increases with prey density but plateaus at high prey densities due to handling time (Holling's disc equation)
• Type III functional response: predation rate initially increases slowly with prey density, then accelerates, and finally plateaus (sigmoidal curve)
• Equilibrium points represent population sizes where predator and prey populations remain constant over time
• Nontrivial equilibrium point occurs when $\frac{dN}{dt} = 0$ and $\frac{dP}{dt} = 0$
• At equilibrium, prey growth balances predator consumption, and predator growth balances mortality

Analysis Techniques

Phase Plane Analysis and Harvesting Models

• Phase plane analysis visualizes the dynamics of predator-prey interactions
• Plots prey population on the x-axis and predator population on the y-axis
• Nullclines represent the lines where $\frac{dN}{dt} = 0$ (prey nullcline) and $\frac{dP}{dt} = 0$ (predator nullcline)
• Intersection of nullclines indicates equilibrium points
• Trajectories in the phase plane show the evolution of predator and prey populations over time
• Stable equilibria occur when trajectories spiral inward towards the equilibrium point (damped oscillations)
• Unstable equilibria occur when trajectories spiral outward away from the equilibrium point (growing oscillations)
• Limit cycles represent closed trajectories in the phase plane, indicating sustained oscillations in predator and prey populations
• Harvesting models incorporate the effects of human exploitation on predator-prey dynamics
• Constant harvesting rate can be added to the Lotka-Volterra equations to analyze the impact of harvesting on population stability and yield
• Optimal harvesting strategies aim to maximize long-term yield while maintaining population sustainability (maximum sustainable yield)