What is the mathematical condition for the deer population to remain constant in the simple model?

Conditions

• The model uses the specific equations $Births = Deer \cdot Birth\_fraction$ and $Deaths = Deer / Lifespan$.
• Parameters are constant over the simulation period.

Reasoning, step by step

1. Set the rate of change of the stock to zero: $d(Deer)/dt = 0$.
2. Substitute the flow equations: $Births - Deaths = 0$.
3. Replace flows with their formulas: $Deer \cdot Birth\_fraction - Deer / Lifespan = 0$.
4. Factor out Deer (assuming Deer > 0): $Birth\_fraction - 1 / Lifespan = 0$.
5. Solve for the relationship: $Birth\_fraction = 1 / Lifespan$.
6. Verify with the example values: $0.2 = 1 / 5$.
7. Conclude that equilibrium is reached when the birth fraction equals the reciprocal of the lifespan.

Example

In the video, when Birth_fraction is 0.2 and Lifespan is 5, the death rate is $1/5 = 0.2$. Since births equal deaths, the graph shows a flat line, indicating a constant population.

Common misconceptions

• Believing the population is constant only if the initial value is zero.
• Thinking that increasing lifespan always leads to growth, ignoring the balance with birth fraction.
• Assuming the stock value itself determines the equilibrium condition rather than the rates.

Watch the explanation

Explore next

• How does increasing the lifespan affect the population trajectory?
• What happens if the birth fraction is lower than 1/lifespan?
• Why does the population grow exponentially when births exceed deaths?