IB MAI HL Calculus Topic 5.4 Differential Equations (id: 5c7b55dd4)

本题目来源于试卷: IB MAI HL Calculus Topic 5.4 Differential Equations，类别为 IB数学

[填空题]
A marine biologist is studying the population growth 1he8j9lj vi h;of a penguin colonjlj/baakifi d23 dc9g. t9/j/ e4 3hbyy in a small archipelago that is free from predators. At the start of her stu9tdj 923/aydi4f /jka ci/.g3ejb bhldy, there was estimated to be 200 penguins in the colony. It is assumed the population growth of penguins can be modelled by the differential equation

$\frac{\mathrm{d} P}{\mathrm{~d} t}$=1.1 P

where P is the penguin population at time t years.
1. Find the population of penguins after 3 years.
After 3 years there are approximately    penguins
​

When the population of penguins reaches 5000 , it is noticed that a group of 100 leopard seals have settled in the area. The subsequent population growth of penguins and leopard seals, where L is the population of leopard seals at time t , can be modelled by the coupled differential equations

$\frac{\mathrm{d} P}{\mathrm{~d} t}$=P(2.2-0.011 L) $\quad \frac{\mathrm{d} L}{\mathrm{~d} t}$=$L(0.0002 P-0.7)$

2. Using Euler's method with a step size of 0.25 , estimate
1. the population of penguins 1 year after the leopard seals were noticed;
After 1 year there are approximately    penguins
2. the population of leopard seals 1 year after they were noticed.
After 1 year there are approximately    leopard seals
​

The graph of the population sizes, according to this model, for the first 4 years after the leopard seals were noticed is shown below.

3. Describe the changes in the populations of penguins and leopard seals for these 4 years,
1. at point A ;
2. at point B .
4. Find the non-zero equilibrium point for the populations of penguins and leopard seals.
therefore the non-zero equilibrium point is
P(  ,  )
​

参考答案:
空格1: 5420  空格2: 11000  空格3: 197  空格4: 3500  空格5: 200 


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