Lecture 1: Population growth and control

A set of practice flashcards covering key concepts from Lecture 1 on population growth, density dependence, equilibria, and related models.

What is carrying capacity (K) in population dynamics?

The maximum number of individuals of a species that the environment can sustain indefinitely without altering the balance of the biocenosis.

What is an attractor in a dynamical system?

The set toward which the system tends to evolve over time from many initial conditions (the long-term behavior of the system).

What is the basin of attraction?

The set of initial conditions in phase space whose trajectories converge to a particular attractor.

What is an equilibrium (fixed point) in population dynamics?

A state where the population growth rate is zero; forces are balanced, and the system may be stable or unstable.

What does r represent in population models and how is it determined?

The intrinsic growth rate, r = b − q; its sign determines whether growth is positive or negative when N is small.

What is the standard logistic growth equation?

dN/dt = rN(1 − N/K), where K is carrying capacity.

What is the Gompertz equation and when does it arise?

A growth model in which the growth rate is proportional to the logarithm of population size; arises when the nonlinearity parameter theta is not zero, leading to growth dependent on log N.

What is the Levins paradox in density-dependent growth?

A concept where, under certain conditions (e.g., N > K and negative per-capita growth), the total population growth can behave counterintuitively.

What is the paradox of r–K strategies?

The idea that increasing the intrinsic growth rate r can be associated with a larger equilibrium carrying capacity K, showing K is not just a fixed ceiling.

In a simple two-variable population–resource model, what are the main state variables and interactions?

N (population size) and R (resource amount); resources flow in, are consumed by the population, and influence carrying capacity through density dependence.

Why might carrying capacity depend on resource dynamics?

Because K is emergent from how resource availability and removal (e.g., mortality or consumption) interact with population growth.

What is the form and interpretation of dN/dt = rN − aN^2?

A density-dependent form where a = r/K; per-capita growth declines linearly with N, and carrying capacity K = r/a.

What does the term 1/N dN/dt = r − aN tell us about population growth?

The per-capita growth rate declines linearly with N, illustrating density dependence leading to a logistic-type slowdown.

Why is the logistic growth graph considered a parabola?

Because dN/dt = rN − (r/K)N^2 is a quadratic function in N, forming a downward-opening parabola with a maximum at intermediate N.

What is the difference between a parabola logistic model and real-world dynamics?

Real dynamics can include paradoxes and deviations (e.g., mortality changes, delayed density dependence, and nonlinearity) beyond the simple logistic form.

What is the effect of adding mortality on equilibrium in density-dependent models (Ginzburg paradox)?

Adding mortality shifts the equilibrium, creating a new effective growth rate r′ and potentially different equilibria.

What is the meaning of a ‘logistic equation is not a logistic function’?

The sigmoidal growth curve is the solution to the logistic differential equation for certain initial conditions, but the underlying function is not itself a pure logistic function.

What is Gompertz growth in relation to theta and log N?

When theta ≠ 0 (near 0), the model can transform into Gompertz form, where growth depends linearly on the log of N.

What is meant by coupled dynamics of resources and population (two-dimension model)?

Resources R and population N interact; carrying capacity depends on resource influx and depletion (aNR term) and mortality/removal processes.

What are Levins’ three properties of models, and what does Levins’ triangle imply?

Generality, realism, and precision; Levins’ triangle states you cannot have all three properties simultaneously in a single model.

What is a projection matrix model, and what does it handle?

A population model (e.g., Leslie matrix) that handles overlapping generations and age/stage structure using a matrix framework.

What did Robert May show about the stability of complex systems?

More complex systems are often less stable; increasing complexity can lead to reduced stability and can produce chaotic dynamics under certain parameters.

What is the logistic map form for a discrete-time population model?

N{t+1} = R Nt (1 − N_t); this can produce fixed points, cycles, or chaos depending on R.

What is the key distinction between determinism and predictability in population dynamics?

A system can be deterministic yet unpredictable due to sensitivity to initial conditions and nonlinear feedbacks.

What is meant by delayed density-dependence and its potential effects?

A time lag between density and its effect on growth, which can produce oscillations and even chaotic dynamics.

What is the main takeaway from the lecture about carrying capacity and population dynamics?

Carrying capacity is the stable equilibrium of the dynamics and can be affected by growth rates, resource dynamics, delays, and other factors that cause oscillations or chaos.