What is the general form of the logistic differential equation?
$\frac{dy}{dt} = ky(M - y)$
Give an alternative form of the logistic differential equation.
$\frac{dy}{dt} = ky(1 - \frac{y}{M})$
How do you calculate the population size when it's growing fastest?
$y = \frac{M}{2}$
What is the carrying capacity (M) when $\frac{dP}{dt} = 0$?
$M = P$
How do you find the carrying capacity (M) from $\frac{dy}{dt} = ky(M - y)$?
M is the value that makes the expression $(M - y)$ equal to zero when y approaches M.
What is the formula for the rate of change of population in a logistic model?
$\frac{dP}{dt} = kP(M - P)$
How do you rewrite $\frac{dP}{dt} = aP(b - cP)$ into the standard logistic form?
$\frac{dP}{dt} = kacP(\frac{b}{c} - P)$
What formula represents the population size when the growth rate is at its maximum?
$P = \frac{M}{2}$, where M is the carrying capacity.
How does the logistic equation relate to exponential growth initially?
When $y$ is much smaller than $M$, the term $(1 - \frac{y}{M})$ is close to 1, and the equation approximates exponential growth: $\frac{dy}{dt} \approx ky$.
What condition must be met to find the carrying capacity?
$\frac{dP}{dt} = 0$
What does the slope of the logistic growth curve represent?
The rate of population growth at that specific time.
How can you identify the carrying capacity on a graph of a logistic model?
The carrying capacity is the horizontal asymptote that the graph approaches as time goes to infinity.
What does the concavity of the logistic growth curve tell you about the population growth?
Concave up means the growth rate is increasing, concave down means the growth rate is decreasing.
How is the point of fastest growth represented on a logistic growth curve?
It is the inflection point, where the concavity changes from up to down.
What does a graph of $\frac{dP}{dt}$ vs. P look like for a logistic model?
It is a downward-facing parabola with roots at P=0 and P=M (carrying capacity), and a vertex at P = M/2.
How can you estimate the carrying capacity from a graph of population vs. time?
Look for the horizontal asymptote, which represents the maximum population size the environment can sustain.
What does the area under the $\frac{dP}{dt}$ curve represent?
The total change in population over a given time interval.
How can you identify the point of maximum growth rate on a graph of $\frac{dP}{dt}$ vs. time?
It is the highest point on the curve, corresponding to the time when the population is growing fastest.
How can you determine if a population is approaching its carrying capacity based on the graph?
The rate of increase in population is decreasing, and the graph is flattening out.
What does a steep slope on a logistic growth graph indicate?
A rapid rate of population growth.
What is a logistic model?
A differential equation describing population growth that slows as it reaches carrying capacity.
Define carrying capacity.
The maximum population size an environment can sustain indefinitely.
What does $\frac{dy}{dt}$ represent in the logistic model?
The rate of change of the population with respect to time.
What does 'k' represent in the logistic equation?
A positive constant representing the growth rate.
What does 'M' represent in the logistic equation?
The carrying capacity of the population.
Define initial population size.
The population size at time t=0.
What is a horizontal asymptote in the context of logistic growth?
A line representing the carrying capacity that the population approaches as time goes to infinity.
What is the significance of the point where population growth is fastest?
It is the point where the rate of change of the population is at its maximum, occurring at half the carrying capacity.
What is the meaning of 'self-limiting' in the context of logistic growth?
It describes a growth process where the rate of growth decreases as the population approaches its carrying capacity.
What is the relationship between carrying capacity and horizontal asymptote?
The carrying capacity is the value of the horizontal asymptote on the graph of a logistic model.
