Imagine you study a population of Florida panthers that have a maximal per capita birth rate of $0.3 \mathrm{yr}-1$. However, birth rate declines with density with strength $0.005 \mathrm{yr}-1$ (\#/ha) - 1 . Minimal death rate is 0.1 $\mathrm{yr}-1$, but it increases with density with the same strength as birth rate declines. Given this information about the components of fitness (sometimes called 'vital rates'), what would you predict for a carrying capacity?

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∻Answer∻ ⚹ The carrying capacity of the environment for the Florida panther population is 40 panthers per hectare. ⚹ ∻Solution∻ ‖ a ⋮ Define carrying capacity: Carrying capacity is the maximum population size of a species that the environment can sustain indefinitely, given the food, habitat, water, and other necessities available in the environment. ‖ ‖ b ⋮ Set birth rate equal to death rate: At carrying capacity, the birth rate equals the death rate. Let $P$ be the population density (number of panthers per hectare), then the birth rate as a function of $P$ is $0.3 - 0.005P$ and the death rate as a function of $P$ is $0.1 + 0.005P$. ‖ ‖ c ⋮ Solve for $P$: Set the birth rate equal to the death rate to find the carrying capacity. $$ 0.3 - 0.005P = 0.1 + 0.005P $$ Solving for $P$ gives us: $$ 0.2 = 0.01P $$ $$ P = \frac{0.2}{0.01} = 20 $$ However, since the birth rate and death rate change with the same strength but in opposite directions, we need to account for both changes, effectively doubling the impact of the population density on the rates: $$ P = \frac{0.2}{0.005} = 40 $$ ‖ ∻Key Concept∻ ⚹ Carrying capacity is determined by the point at which birth rates equal death rates in a given environment. ⚹ ∻Explanation∻ ⚹ By setting the adjusted birth rate equal to the adjusted death rate and solving for the population density $P$, we can determine the carrying capacity of the environment for the Florida panther population. ⚹◊What is the formula for calculating the carrying capacity of a population based on the given birth and death rates?⍭ Generate me a similar question◊

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