Let's plug in some specific numbers for our constants: HP = 0.4H – 0.01H L dt TP –0.3L+0.005H L dt One useful thing to know is if the population could ever be stable. Are there values of H and L for which neither population changes? What equations would we want to solve to find such values?
Let's plug in some specific numbers for our constants: HP = 0.4H – 0.01H L dt TP –0.3L+0.005H L dt One useful thing to know is if the population could ever be stable. Are there values of H and L for which neither population changes? What equations would we want to solve to find such values?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Equations and inequalities describe the relationship between two mathematical expressions.
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A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
L = Lynx
H = Hares
t = time
- also solve the equations. Two sets of values (H,L) should be found.
![6. Let's plug in some specific numbers for our constants:
\[
\frac{dH}{dt} = 0.4H - 0.01HL
\]
\[
\frac{dL}{dt} = -0.3L + 0.005HL
\]
One useful thing to know is if the population could ever be stable. Are there values of \( H \) and \( L \) for which neither population changes? What equations would we want to solve to find such values?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4e3da8d7-5e45-4b35-92f7-ecdb5bd91ee7%2F1742cdaf-aa60-4634-92c5-561d8a7fbc9c%2Ffwswp74_processed.jpeg&w=3840&q=75)
Transcribed Image Text:6. Let's plug in some specific numbers for our constants:
\[
\frac{dH}{dt} = 0.4H - 0.01HL
\]
\[
\frac{dL}{dt} = -0.3L + 0.005HL
\]
One useful thing to know is if the population could ever be stable. Are there values of \( H \) and \( L \) for which neither population changes? What equations would we want to solve to find such values?
![5. Okay, time for some mathematics. We're going to focus on the interactions between the hares and their main predators, the lynx. The tool we need is differential equations (last seen when we talked about the S-I-R disease model). We will let \( H(t) \) be the density of hares at time \( t \), and \( L(t) \) the density of lynx at time \( t \).
Our first assumption is that, if there are no lynx around, the hares grow exponentially. Hares lead to more hares and faster growth. This gives us the differential equation (trust me):
\[
\frac{dH}{dt} = aH
\]
where \( a \) is a positive constant.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4e3da8d7-5e45-4b35-92f7-ecdb5bd91ee7%2F1742cdaf-aa60-4634-92c5-561d8a7fbc9c%2Fqgycmfi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:5. Okay, time for some mathematics. We're going to focus on the interactions between the hares and their main predators, the lynx. The tool we need is differential equations (last seen when we talked about the S-I-R disease model). We will let \( H(t) \) be the density of hares at time \( t \), and \( L(t) \) the density of lynx at time \( t \).
Our first assumption is that, if there are no lynx around, the hares grow exponentially. Hares lead to more hares and faster growth. This gives us the differential equation (trust me):
\[
\frac{dH}{dt} = aH
\]
where \( a \) is a positive constant.
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