4.4 The logistic equation  (Page 6/12)

[T] For the previous fishing problem, draw a directional field assuming $k=0.1.$ Draw some solutions that exhibit this behavior. What are the equilibria and what are their stabilities?

[T] Use software or a calculator to draw directional fields for $k=0.4.$ What are the nonnegative equilibria and their stabilities?

[T] Use software or a calculator to draw directional fields for $k=0.6.$ What are the equilibria and their stabilities?

Solve this equation, assuming a value of $k=0.05$ and an initial condition of $2000$ fish.

Solve this equation, assuming a value of $k=0.05$ and an initial condition of $5000$ fish.

Draw the directional field of the threshold logistic equation, assuming $K=10,r=0.1,T=2.$ When does the population survive? When does it go extinct?

For the preceding problem, solve the logistic threshold equation, assuming the initial condition $P\left(0\right)=P_0.$

Bengal tigers in a conservation park have a carrying capacity of $100$ and need a minimum of $10$ to survive. If they grow in population at a rate of $1\text{\%}$ per year, with an initial population of $15$ tigers, solve for the number of tigers present.

A forest containing ring-tailed lemurs in Madagascar has the potential to support $5000$ individuals, and the lemur population grows at a rate of $5\text{\%}$ per year. A minimum of $500$ individuals is needed for the lemurs to survive. Given an initial population of $600$ lemurs, solve for the population of lemurs.

The population of mountain lions in Northern Arizona has an estimated carrying capacity of $250$ and grows at a rate of $0.25\text{\%}$ per year and there must be $25$ for the population to survive. With an initial population of $30$ mountain lions, how many years will it take to get the mountain lions off the endangered species list (at least $100)?$

The Gompertz equation is given by $P\left(t\right)\prime =\alpha \text{ln}\left(\frac{K}{P\left(t\right)}\right)P\left(t\right).$ Draw the directional fields for this equation assuming all parameters are positive, and given that $K=1.$

Assume that for a population, $K=1000$ and $\alpha =0.05.$ Draw the directional field associated with this differential equation and draw a few solutions. What is the behavior of the population?

Solve the Gompertz equation for generic $\alpha$ and $K$ and $P\left(0\right)=P_0.$

[T] The Gompertz equation has been used to model tumor growth in the human body. Starting from one tumor cell on day $1$ and assuming $\alpha =0.1$ and a carrying capacity of $10$ million cells, how long does it take to reach “detection” stage at $5$ million cells?

[T] It is estimated that the world human population reached $3$ billion people in $1959$ and $6$ billion in $1999.$ Assuming a carrying capacity of $16$ billion humans, write and solve the differential equation for logistic growth, and determine what year the population reached $7$ billion.

[T] It is estimated that the world human population reached $3$ billion people in $1959$ and $6$ billion in $1999.$ Assuming a carrying capacity of $16$ billion humans, write and solve the differential equation for Gompertz growth, and determine what year the population reached $7$ billion. Was logistic growth or Gompertz growth more accurate, considering world population reached $7$ billion on October $31,2011?$

Show that the population grows fastest when it reaches half the carrying capacity for the logistic equation $P\prime =rP\left(1-\frac{P}{K}\right).$

When does population increase the fastest in the threshold logistic equation $P\prime \left(t\right)=rP\left(1-\frac{P}{K}\right)\left(1-\frac{T}{P}\right)?$

When does population increase the fastest for the Gompertz equation $P\left(t\right)\prime =\alpha \text{ln}\left(\frac{K}{P\left(t\right)}\right)P\left(t\right)?$

Find the equation and parameter $r$ that best fit the data for the logistic equation.

Find the equation and parameters $r$ and $T$ that best fit the data for the threshold logistic equation.

Find the equation and parameter $\alpha$ that best fit the data for the Gompertz equation.

Graph all three solutions and the data on the same graph. Which model appears to be most accurate?

Using the three equations found in the previous problems, estimate the population in $2010$ (year $70$ after conservation). The real population measured at that time was $437.$ Which model is most accurate?