math modeling The population of three species X , Y and Z is governed by the sys
ID: 3108403 • Letter: M
Question
math modeling
The population of three species X , Y and Z is governed by the system dX/dt = X + Y , dY/dt = a X + 2Y 3Z , dZ/dt = Y + 3Z where a is a positive constant.
(a) Which of the following interactions is best modeled by the system (explain your reasoning)
(i) X eats Y and Y eats Z ; (ii) X eats Y and Y , Z compete for food;
(iii) X , Y compete for food and Y eats Z ; (iv) X , Y and Z all compete for food.
(b) Find the value of a for which there is a nontrivial equilibrium population distribution.
(c) If the total population in (b) is 14,000, how many individuals are there of X , Y and Z ?
Explanation / Answer
dX/dt = X + Y , dY/dt = a X + 2Y 3Z , dZ/dt = Y + 3Z
i) X eats Y and Y eats Z
Because the working convention is that "positive" is not the same as "not negative".
To sketch the solution curves, the easy solution is to use a computer program that can solve ordinary differential equations numerically. (I generally use Maple, but there are plenty of other tools that can do it. Even a good graphing calculator should be enough.)
Alternatively, you can make pretty good sketches by hand just by remembering that near stable equilibria the orbits will generally decay exponentially closer to the equilibrium, while near unstable equilibria they will grow exponentially away from it. When you have an orbit starting near an unstable equilibrium and ending near a stable one, the ends will connect to form a sigmoid shape. The details will depend on the exact form of the growth rate function, but they don't affect the general rule that orbits move away from unstable equilibria and towards stable ones.
Ps. I deliberately avoided showing any of the actual calculations in my answer. Let me know if you need more help with something — or, better yet, show us what you've managed to do so far, so that we can help if you get stuck on a particular detail.
(There are other discrepancies between countries about terminology, which later were spread through the rest of the world.)
Because the working convention is that "positive" is not the same as "not negative".
To sketch the solution curves, the easy solution is to use a computer program that can solve ordinary differential equations numerically. (I generally use Maple, but there are plenty of other tools that can do it. Even a good graphing calculator should be enough.)
Alternatively, you can make pretty good sketches by hand just by remembering that near stable equilibria the orbits will generally decay exponentially closer to the equilibrium, while near unstable equilibria they will grow exponentially away from it. When you have an orbit starting near an unstable equilibrium and ending near a stable one, the ends will connect to form a sigmoid shape. The details will depend on the exact form of the growth rate function, but they don't affect the general rule that orbits move away from unstable equilibria and towards stable ones.
Ps. I deliberately avoided showing any of the actual calculations in my answer. Let me know if you need more help with something — or, better yet, show us what you've managed to do so far, so that we can help if you get stuck on a particular detail.
(There are other discrepancies between countries about terminology, which later were spread through the rest of the world.)
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