Source of story and modeling idea: Mathematical Modelling with Case Studies, by Belinda Barnes and Glenn Fulford.
Overall objectives:
In this lab we will build upon the concepts developed in our recent lab on
rate of change models for different growth processes. This time we'll conjecture and design a model for the rate of
change of two populations that interact with each other -- the Australian scale-insect (prey) and the beetle (predator).
We make the following two assumptions which, though inaccurate, lead to remarkably good real-world models:
(I) Start with simple case - assume each population is on its own
(there is no interaction between them)
\n\n\nSource of story and modeling idea: Mathematical Modelling with Case Studies, by Belinda Barnes and Glenn Fulford.\n\n\n
\n\nOverall objectives:
\nIn this lab we will build upon the concepts developed in our recent lab on \nrate of change models for different growth processes. This time we'll conjecture and design a model for the rate of \nchange of two populations that interact with each other -- the Australian scale-insect (prey) and the beetle (predator). \n
\n\nWe make the following two assumptions which, though inaccurate, lead to remarkably good real-world models:\n\n
\n(I) Start with simple case - assume each population is on its own
\n(there is no interaction between them)\n
Some examples:
$S^\prime=1$, OR $~S^\prime=t$, OR $~S^\prime=e^t$, OR $~S^\prime=S$, OR ...
Clearly each of these corresponds to an increasing function $S(t)$. [Why?]
In that sense, each is a plausible model for the scale-insect population.
But, which of them does the "best" job? Why?
Write an analogous model for $B^\prime$.
\nSome examples:
\n$S^\\prime=1$, OR $~S^\\prime=t$, OR $~S^\\prime=e^t$, OR $~S^\\prime=S$, OR ...
\nClearly each of these corresponds to an increasing function $S(t)$. [Why?]
\nIn that sense, each is a plausible model for the scale-insect population.
\nBut, which of them does the \"best\" job? Why?
\nWrite an analogous model for $B^\\prime$.\n
Things to remember: $S^\prime(t)$ = negative term, causes $S(t)$ to decrease;
S$^\prime(t)$ = positive term, causes $S(t)$ to increase;
$S^\prime(t)$ = negative + positive term, causes $S(t)$ to respond to two opposing effects.
Here are some explorations to try:
We'll carry out the simulation for 3 different starting
scenarios:
(i) At t=0 there are 500 scale-insects
and 50 beetles.
(ii) At t=0 there are 500 scale-insects
and 100 beetles.
(iii) At t=0 there are 1000 scale-insects
and 81 beetles.
Consult with Anand for help on how to use Sage
to solve this model.
Discuss what your simulations reveal. In particular, discuss what types of patterns you see in the predator and prey populations, and whether/ how they depend on the initial population of each species. For each simulation, use your graphical solutions to estimate an average value of the predator and prey populations. A very rough approximation of the averages is sufficient.
Run the grapher simulation again after adding the following
new terms to your previous model:
S'(t) = previous terms - 0.01 S.
B'(t) = previous terms - 0.01 B.
Comment on what your results reveal about the effect of using DDT. Address any changes you see in the population patterns, and in the average values. ︡44576692-86a4-44c9-841f-50106e937510︡{"hide":"input"}︡{"html":"(II) More realistic (& interesting) case - the predator and prey populations interact
\n
\nThings to remember: $S^\\prime(t)$ = negative term, causes $S(t)$ to decrease;
S$^\\prime(t)$ = positive term, causes $S(t)$ to increase;
\n$S^\\prime(t)$ = negative + positive term, causes $S(t)$ to respond to two opposing effects.
\n\nHere are some explorations to try:
\n\n
\nWe'll carry out the simulation for 3 different starting \nscenarios:
\n (i) At t=0 there are 500 scale-insects \nand 50 beetles.
\n (ii) At t=0 there are 500 scale-insects \nand 100 beetles.
\n (iii) At t=0 there are 1000 scale-insects \nand 81 beetles.
\nConsult with Anand for help on how to use Sage \nto solve this model.
\n\nDiscuss \nwhat your simulations reveal. \nIn particular, discuss what types of patterns you see \nin the predator and prey populations, and whether/ how\nthey depend on the initial population of each species. \nFor each simulation, \nuse your graphical solutions to estimate an average value \nof the predator and prey populations. A very rough \napproximation of the averages is sufficient.\n\n
\n\nRun the grapher simulation again after adding the following \nnew terms to your previous model:
\n S'(t) = previous terms - 0.01 S.
\n B'(t) = previous terms - 0.01 B.
\n\nComment on what your results reveal about the effect of \nusing DDT. Address any \nchanges you see in the population patterns, and in the average \nvalues.\n\n"}︡{"done":true}︡ ︠d6283df0-2eea-4cef-a2b2-11a434ceb846︠ # Systems is: S'= 0.08*S - 0.001*S*B; B'= -0.02*B + 0.00002*S*B; S(0)=500, B(0)=50 s, b, t = var('s b t') de1 = 0.08*s - 0.001*s*b de2 = -0.02*b + 0.00002*s*b P = desolve_system_rk4 ([de1, de2], [s, b], ics=[0, 500, 50], ivar=t, end_points=[0,200] ) Q = [ [j,k] for i, j, k in P] #list_plot(Q, axes_labels=['$S$', '$B$']) line(Q, axes_labels=['$S$', '$B$'], color='green', thickness=2, title='Phase plot') # The following creates plots of S vs t and B vs t: QS = [ [i,j] for i, j, k in P] QB = [ [i,k] for i, j, k in P] SS = line(QS, axes_labels=['$t$ (days)', '$P(t)$'], legend_label='Scale insects', color='blue', thickness=2) SB = line(QB, axes_labels=['$t$ (days)', '$P(t)$'], legend_label='Beetles', color='green', thickness=2) show(SS+SB) ︡b04387c3-9e9f-4d7e-9903-9a058eef942e︡{"file":{"filename":"/projects/c5fbbb89-591d-4df5-a627-cb7020bf6ed6/.sage/temp/compute2-us/28003/tmp_K7v7Do.svg","show":true,"text":null,"uuid":"98276a83-9fe0-4ec8-82ab-7dea412c4fcb"},"once":false}︡{"file":{"filename":"/projects/c5fbbb89-591d-4df5-a627-cb7020bf6ed6/.sage/temp/compute2-us/28003/tmp_7LDVbp.svg","show":true,"text":null,"uuid":"085840e0-9465-4e81-842c-c4fd8f95696d"},"once":false}︡{"done":true}︡