What is Evolution?

Key Concepts

  • Evolution:

    • Refers to changes in the frequency of different types within a population over generations.
    • In biological terms, it specifically describes changes in allele frequencies in a gene pool.

  • Reproduction:

    • Evolution requires a population of reproducing individuals for allele frequencies to evolve.

  • Exponential Growth ("Malthusian law"):

    • Discrete Model: Population doubles with each generation. The growth equation is \( x_{t+1} = 2x_t \), leading to \( x_t = x_0 \cdot 2^t \).
    • Continuous Model: For continuous time, the growth rate is described by \( x'(t) = r \cdot x(t) \) with solution \( x(t) = x_0 e^{rt} \).

  • Cell Death:

    • Mortality is included in the population model with the equation \( x'(t) = (r - d) \cdot x(t) \), where \(d\) is the death rate.
    • The basic reproductive ratio \( R_0 = \frac{r}{d} \) defines population growth dynamics:
      • \( R_0 > 1 \): Population grows.
      • \( R_0 < 1 \): Population shrinks to extinction.
      • \( R_0 = 1 \): Population size remains constant but is unstable.

  • Logistic Growth:

    • When a population approaches its carrying capacity \(K\), the growth rate slows according to the logistic equation: \[ x'(t) = r x(t) \left( 1 - \frac{x(t)}{K} \right) \]
    • The population reaches equilibrium at the carrying capacity \(K\).

  • Logistic Difference Equation:

    • In discrete time, the logistic map is described by \( x_{t+1} = a x_t (1 - x_t) \), where \(a\) is the growth rate.
    • This simple equation can exhibit chaotic behavior for certain values of \(a\).

  • Selection:

    • Independent Types: For two types \(A\) and \(B\) with exponential growth rates \(a\) and \(b\), their relative proportions evolve according to: \[ \frac{x(t)}{y(t)} = \frac{x_0}{y_0} e^{(a - b)t} \]
      • If \(a > b\), type \(A\) outcompetes type \(B\), and vice versa.

  • Two Competing Types:

    • When the total population is constrained, the selection dynamics for two competing types \(A\) and \(B\) follow: \[ x'(t) = x(t)(1 - x(t))(a - b) \]
    • The system describes the competition between two types based on their fitness.

  • Probability Simplex:

    • The state of a population with multiple types is represented on a probability simplex, where each point corresponds to the relative frequencies of types in a population.

  • Subexponential vs. Superexponential Growth:

    • Subexponential Growth (\(c < 1\)): Stable coexistence of multiple types, even if one has a fitness advantage.
    • Superexponential Growth (\(c > 1\)): One type dominates and drives the other to extinction, leading to an unstable mixed equilibrium.

  • Mutation:

    • Mutation introduces genetic diversity, even in the absence of selection.
    • For two types with mutation rates \(u_1\) and \(u_2\), the equilibrium frequencies depend on the mutation rates: \[ x^* = \frac{u_2}{u_1 + u_2} \]
      • Mutation allows coexistence of types, even with no fitness differences.

  • Hardy-Weinberg Principle:

    • Describes how allele frequencies in a large, randomly mating population reach equilibrium after one round of random mating.
    • Genotype frequencies follow: \[ x = p^2, \quad y = 2pq, \quad z = q^2 \]
    • These frequencies remain constant over generations unless external factors such as mutation or selection are introduced.

Key Equations and Models

  • Exponential Growth: \[ x'(t) = r \cdot x(t) \quad \Rightarrow \quad x(t) = x_0 e^{rt} \]

    • Describes exponential population growth in continuous time.

  • Logistic Growth: \[ x'(t) = r \cdot x(t) \left( 1 - \frac{x(t)}{K} \right) \quad \Rightarrow \quad x(t) = \frac{Kx_0 e^{rt}}{K + x_0 (e^{rt} - 1)} \]

    • Describes population growth with carrying capacity \(K\).

  • Selection Dynamics: \[ x'(t) = x(t)(1 - x(t))(a - b) \]

    • Models competition between two types based on relative fitness.

  • Mutation Dynamics: \[ x'(t) = u_2 - x(u_1 + u_2) \quad \Rightarrow \quad x^* = \frac{u_2}{u_1 + u_2} \]

    • Describes coexistence due to mutation, even when both types have equal fitness.

  • Hardy-Weinberg Equilibrium: \[ p = x + \frac{y}{2}, \quad q = z + \frac{y}{2} \]

    • Allele frequencies remain constant under random mating.

Summary Points

  • Evolution is driven by the reproduction of individuals and changes in allele frequencies over generations.
  • Population growth can be modeled using exponential and logistic equations, which describe different growth dynamics, including constraints like death and carrying capacity.
  • Selection dynamics favor individuals with higher relative fitness, leading to the survival of the fittest.
  • Subexponential growth allows for coexistence, while superexponential growth leads to the dominance of one type.
  • Mutation introduces genetic variation and can drive coexistence even in the absence of fitness differences.
  • The Hardy-Weinberg principle explains how allele and genotype frequencies stabilize in a large, randomly mating population.