Question

Bison in Yellowstone National Park have a food-limited carrying capacity that determines how their population grows. If the population grows too large, some bison will either starve or migrate to search for more food. The park is able to support a maximum population of 4,500 bison. Park conservation officials must watch the population and prevent the bison from overpopulating, since they would migrate to nearby farms, causing agricultural damage and creating hazards to drivers. The bison begin to migrate when the population reaches 4,000 because of competition for food. If the current population size of bison is 3,652 and the maximum growth rate of the population is 0.28, calculate the population size after one year and determine whether the park will need to take measures to control the population.

A. The population will be 3,845 after a year. The bison will not pose a problem until the following year, and no conservation action will need to take place.
B. The population will be 4,675 after a year. Some bison will need to be removed to prevent migration, since they will surpass the resource availability.
C. The population will be 1,023 after a year, since the growth rate is too low. Conservation efforts will need to focus on increasing the bison population.
D. The population will be 4,500 after a year. The bison cannot exceed the carrying capacity, so they will stop reproducing. Some bison will need to be removed.

Answer 1

A. The population will be 3,845 after a year. The bison will not pose a problem until the following year, and no conservation action will need to take place.

Step-by-step explanation:

The exposed example refers to a population that exhibits a logistic growth model, Where the principal limiting factor is food availability

• N is the total size of the population
• Nt the population size after a T time
• K the carrying capacity
• r the population growth rate per capita

When N < K the population has not reached the maximum size that can be supported by the environment. It can increase in size.

When N = K the population has reached the maximum size that can be supported by the environment. It can not grow anymore. The population reached a maximum point delimited by available resources, such as food

When N > K the population has supered the maximum size that can be supported by the environment. It must decrease in size. because there are not enough resources.

To calculate the population size after one year, we need to use the following equation.

N t+1= Nt+rN(1-N/K)

Available data:

K = 4500

N = 3652

r = 0.28

Nm= 4000

N/K = 3652/4500 = 0.8115

r.N = 0.28 x 3652 = 1022.56

N t+1= Nt+rN(1-N/K)

N t+1= 3652+1022.56(1-0.8115)

N t+1= 3652+1022.56(0.1885)

N t+1= 3652+192.753

N t+1= 3844.75

rounding up to 3845