Question

Mind to help Checking my Answer? Thanks~
A population of a rare insect in an area will grow at a rate that is proportional to their current population. In the absence of any outside factors, the population will triple in two weeks’ time. On any given day there is a net migration (into the area) of 15 insects and there are also an extinction cause by identified factor such as 16 are eaten by the local bird population and 7 die of natural causes. If there are initially 100 insects in the area, predict either the population of the rare insect will survive or not? If not, when do they will die out

Answer 1

is it right for me to state that \[\frac{ dx }{ dt }=x(\frac{ 3 }{ 14 })-8 ?\]

Answer 2

I then get \[dt=\frac{ 14 }{ 3 } ( \frac{ dx }{ x - \frac{ 112 }{ 3 } } )\]\[t + c= \frac{ 14 }{ 3 } \ln (x - \frac{ 112 }{ 3 } )\]\[\frac{ 3 }{ 14 }t + C= \ln(x-\frac{ 112 }{ 3 } )\]\[x - \frac{ 112 }{ 3 } = e^{\frac{ 3 }{ 14 }} C\]After that I sub the given value and get C, and sub x=0 to get t.
But I'm wondering, am I correct?

Answer 3

I'm not sure am I right about the rate of increasing, which I put as 3/14

Answer 4

I don't know how to construct the equation, but I know how to solve it.

Answer 5

please help me :)

Answer 6

If I give you an invisible cupcake will you give me a medal?

Answer 7

I'm willing to give medal to someone who can help me out :)

Answer 8

Must your approach involve a differential equation? I notice that this is inherently a discrete situation and that the numbers involved are small.

I can choose a rate of increase so that the population triples in 14 steps:

0 107.597
1 115.77114409
2 124.566277907
3 134.029578039
4 144.211805083
5 155.167575915
6 166.955656657
7 179.639277893
8 193.286473835
9 207.970447252
10 223.76996213
11 240.769766153
12 259.061045288
13 278.741912898
14 299.917936021

Then I can decrement the increase at each step by 8 to see what effect that has on the population size.

0 99.597
1 99.16338409
2 98.6968263793
3 98.1948242794
4 97.6546850799
5 97.0735115054
6 96.4481861744
7 95.7753548781
8 95.0514085882
9 94.2724640986
10 93.4343431962
11 92.5325502488
12 91.5622480912
13 90.5182320787
14 89.3949021698

Clearly the population is dying out.

Answer 9

Thanks for the respond, so the population is an exponential growth? I thought the absence of any outer factor will make it a directly proportional growth, which is 3/14

Answer 10

what is your C?

Answer 11

I dint calculate out the C, but I think we can get it by substitute x= 100 and t=0?

Answer 12

The question says that population growth rate is proportional to population size, and that was reflected in your original work. As I see it, the difficulty with this problem is that counts are taken in discrete time yet you might prefer to use differential equations which are essentially continuous.

Answer 13

I think I can see the picture already, thanks so much :)

Answer 14

You're most welcome.