Week 2: Density Dependent Population Growth

Why is population growth density-dependent?

• populations don’t grow forever

• limited by resources, competitors, natural enemies

• Intraspecific competition

  • Exploitation

  • Interference→ individuals prevent access to resources e.g fighting over food

• Vital rates change according to conspecific density

  • Negative effects strong at high densities

  • Often positive effects at low densities

As density increases, what happens to survival rate/ individual fecundity?

decreases

Density dependent responses in the life stages of plants

• Seed predation/granivory

• Seed germination

• Seedling establishment/survival

• Sapling/tree/adult growth & survival

• Seed production

  • pollen availability

  • size of adult

What is the (Warder Clyde) Allee effect?

• aggregation can improve the survival rate of individuals, and that cooperation may be crucial in the overall evolution of social structure

Example of the Allee effect

• Mating success

• Defence→ safety in numbers

• Feeding e.g Yellowstone wolves hunt together - increasing pack size increases p(attack) → p(capturing) - both increase as pack size increases - but there is a peak and it decreases after that point

When is the Allee effect strong?

• at low pop. density + low growth rat

• to do with mating success

What is the per capita growth rate?

number next year/ number in current year

Behaviour that affects density-dependent growth

• Dispersal

  • avoid competition

  • avoid kin

  • avoid poor resource levels

• Feeding

  • interference competition

  • ‘arguing’ rather than feeding

  • or more cooperation

How are movement and dispersal different?

• Movement eg leaving a food patch

• Dispersal –normally leads to novel gene transfer (mating)

How is movement + dispersal density dependent?

• Positive density dependent

  • Avoid competition now

  • Find new resources

• Negative density dependent

  • Avoid competition later(?)

  • Mate finding

What are the 3 types of negative density dependence?

1. Under-compensatory density dependence

   • Causes mortality

   • Births still greater than deaths

2. Compensatory density dependence

   • Deaths = births

3. Over-compensatory density dependence

   • Fewer individuals next time step than current time step

What is the self-thinning rule?

• The self-thinning rule relates plant mass to plant density in crowded, even-aged stands by a power-law equation with an exponent −3/2.

What kind of organisms does the law of self-thinning apply to?

• sessile

• especially plants

How does density affect the size/ biomass of individuals

Increase in density = reduction in size

Decrease in density = increase in size

-3/2 slope

• 3/2 means weight increases faster than numbers decreasing → cannot continue forever

• At some point weight must increase at same rate

  • Slope = -1

  • Lower light → shows slope -1

• Some variation in slope

  • Indicative of how close population is to limit (max yield)

Carrying capacity (K)

The maximum sustainable population size in the current environment

Population growth phases

• Small populations tend to grow exponentially= Density independent phase→ simple equation from last week

• Population growth slows as environmental carrying capacity is approached

Equilibria

growth of population is zero

What is a phase plane (in a time series of pop. dynamics)?

density now vs. density in next time point

What 2 kinds of phase planes can you have?

• 2 species (N₁ vs. N₂)

• 1 species (N_t vs. N_t+1)

What are attractors?

• Set of points in phase space.

• Once reached, system does not leave set.

• All points in set are reached.

• Any point in phase space near attractor will approach the attractor

What are the 3 types of attractors?

• Stable (point)

• Limit cycle (series of points/ ring)

• Chaos (complex geometry)

Damped oscillations

overshoot, undershoot, gets smaller → eventually reaches stable equilibrium

2-point limit cycle

• In equilibrium (not changing its behaviour over time)→ balanced between one value overshooting and one undershooting

• Purely driven by density dependence

What is Chaos?

• Extreme sensitivity to initial conditions (noise) and non-periodic

• Not random/ stochastic

• Deterministic: IF exact initial conditions known then complete prediction possible

• Due to over-compensatory density dependence

What does chaos look like on a graph?

• ^ looks random but is not → parameters stay the same so nothing changes in the model over time

• only predictable thing is that it goes up and down

• gets very close to 0

• never repeats itself → never finds a sequence that repeats

• still at an equilibrium

• limit on max. number it will go to

• chaos→ slightly different starting conditions → different trajectories = sensitive

Non-overlapping generations

• discrete time

• difference equations

• e.g insects

Overlapping generations

e.g perennial plants, humans

• continuous time

• differential equations

Equation to model density-dependence growth

Ricker equation

dN/ dt

• RATE OF CHANGE of N with respect to t

• gradient

• instantaneous rate of change

dN/ dt < 0

population in decline

dN/ dt < 0

population increasing

dN/ dt = 0

population in equilibrium

Properties of the logistic population model

• S-like population growth curve.

• Max growth rate at 1/2 carrying capacity.

• Large populations take longest to reach equilibrium.

• Stable equilibrium (chaos not possible) -point attractor