**Activity 52.2 What Models Can You Use to Calculate How Quickly a Population Can Grow?**
1. In the simplest population growth model (*dN*/*dt* *rN*).
a. What do each of the terms stand for?
**Term Stands for: **
*dN *Change in the number of individuals
*dt *Time interval during which the change occurred
*r * Per capita population growth rate *b – d*
*N *Initial population size
b. What type of population growth does this equation describe?
Exponential growth of a population (J-shaped curve)
c. What assumptions are made to develop this equation?
Unlimited space (habitat) and resources
2. Population growth may also be represented by the model, *dN*/*dt* *rN*[(*K* – *N*)/*K*].
a. What is *K*?
Carrying capacity, or the number of individuals that can be sustained over time. *K* is a function of the environment.
b. If *N* *K*, then what is *dN*/*dt*?
*dN/dt* 0; that is, there is no change in the population size over time.
c. Describe in words how *dN*/*dt* changes from when *N* is very small to when *N* is large relative to *K*.
When *N* is very small, the population is growing exponentially. The actual number of new individuals added is relatively small, however, because* N* is so small. As *N* nears the value of *K*/2, *dN/dt* approaches its maximum. As *N* approaches *K*, *dN/dt* decreases.
d. What assumptions are made to develop this equation?
The logistic model assumes that resources and space limit the growth of a population and that these factors determine the maximum number of individuals that can be sustained over time.
POPULATION GROWTH PROBLEMS
Answers:
1) r is increasing. r=0.045
2) 172 frogs
3). dN/dt= (b-d)N
(1134-1492)/1 = (b- 0.395) 1492
so b= 0.155.
Not suitable, because birth rate is much lower than death rate.
4a) 8.5 turkeys/acre
4b) 110 turkeys/ year
4c) 890 turkeys
5) One dandelion plant can produce many seeds, leading to a high growth rate for dandelion populations. If a population of dandelions is currently 40 individuals, and r_{max}= 80 dandelions/month, predict dN/dt if these dandelions would grow exponentially.
Equation to use, exponential growth: dN/dt= r_{max} N
dN/dt= 80 x 40= 3200
3200 dandelions
6) Imagine the dandelions mentioned in #5 cannot grow exponentially, due to lack of space. The carrying capacity for their patch of lawn is 70 dandelions. What is their dN/dt in this logistic growth situation?
Equation to use, logistic growth: dN/dt= r max N (K-N/K)
dN/dt= 80 x 40 (70-40/70)
dN/dt= 3200 (30/70)
dN/dt= 1371
1371 dandelions
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