Matrix models

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# this can not be made into a procedure of a list because of a bug in Maple; see bug.mw

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

#temppopP(P,u,T) = tempplot([orbitP(P,u,T)])

#temppopF(P,u,T) = tempplot([orbitF(F,u,T)])

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We consider the matrix model from Allman and Rhodes, Mathematical models in biology, p 42, for an insect population.

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The 20-generation population orbit through u0=<0,0,5> is

QyQ+SSJTRzYiLUknb3JiaXRQR0YlNiUmSSJQR0YlNiNJJ2luc2VjdEdGJS1JJDwsPkdGJTYlIiIhRjAiIiYiIz8hIiI=

QyQtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSSQ8LD5HSShfc3lzbGliRzYiNiMtRiQ2JC1JJmZsb29yRzYkRiVGKTYjJiZJIlNHRio2I0kiakdGKjYjSSJpR0YqL0Y4OyIiIiIiJC9GNjtGOyIjP0Y7

The temporal graph of the population vectors is

QyQtSSl0ZW1wcGxvdEc2IjYjSSJTR0YlIiIi

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QyQtSSdvcmJpdEZHNiI2JWYqNiNJInVHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJS1JIk5HRiU2JCZJIlBHRiU2I0knaW5zZWN0R0YlOSRGJUYlRiUtSSRucm1HRiU2Iy1JJDwsPkdJKF9zeXNsaWJHRiU2JSIiIUY8IiImIiM/ISIi

QyQ+JkkjUERHNiI2I0knaW5zZWN0R0YmLUkkc2VxRyUqcHJvdGVjdGVkRzYkLUkkPCw+R0koX3N5c2xpYkdGJjYjLUYqNiQtSSZmbG9vckc2JEYrRi82IyYmSSIlR0YmNiNJImpHRiY2I0kiaUdGJi9GPTsiIiIiIiQvRjs7RkAiIz9GQA==

stabilizes.

Here is a graph of the population distribution

QyQtSSl0ZW1wcGxvdEc2IjYjNyMmSSNQREdGJTYjSSdpbnNlY3RHRiUiIiI=

The stable population distribution is

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The total population through the 20 generations

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grows (approximately) exponentially: The total populations size is c* \316\273_\342\210\236^t for some constant c and some stable growth rate \316\273_\342\210\236.

Conclusion

1.The population distribution stabilizes at the stable population distribution

2. The total population grows exponentially with a stable growth rate

Let us now try to determine the stable growth rate \316\273_\342\210\236.

Since y_\342\210\236 is a stable distribution, we expect Py_\342\210\236 to have the same population distribution. This means that we expect Py_\342\210\236 to be proportional to y_\342\210\236. The proportionality factor must be the stable growth rate \316\273_\342\210\236.

We therefore estimate the stable growth rate to be

QyQ+JkknbGFtYmRhRzYiNiNJKWluZmluaXR5RyUqcHJvdGVjdGVkRyomLUkkYWRkR0YpNiQmLUkwZGVsYXlEb3RQcm9kdWN0RzYkRikvSSttb2R1bGVuYW1lR0YmSSxUeXBlc2V0dGluZ0c2JEYpSShfc3lzbGliR0YmNiQmSSJQR0YmNiNJJ2luc2VjdEdGJiZJInlHRiZGJzYjSSJpR0YmL0Y/OyIiIiIiJEZCLUYsNiQmRjxGPkZAISIiRkI=

We choose the constant c so that the two graphs agree at t=20, so that total population size at t=20 is c* \316\273_\342\210\236^(20).

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Here is the predicted exponential growth of the total population compared to the actual total population

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We see a good match between the two curves.

Based on this, we predict that the population vector is approximately c/add(y_\342\210\236[i],i=1..3)*\316\273_\342\210\236^t*y_\342\210\236.

We compare this prediction to the actual population vector

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The Fundamental Theorem of Demography says that the stable growth rate is the dominating eigenvalue and the stable distribution is its eigenvector.

We check that the stable growth rate is the dominating eigenvalue.

The eigenvalues are

QyQ+SSplaWdlbnZhbHNHNiItSSxFaWdlbnZhbHVlc0dGJTYjJkkiUEdGJTYjSSdpbnNlY3RHRiUiIiI=

so the dominating eigenvalue is

QyQ+SSJsRzYiLUkkbWF4RyUqcHJvdGVjdGVkRzYjLUkkc2VxR0YoNiQtSSRhYnNHRig2I0knbGFtYmRhR0YlL0YwSSplaWdlbnZhbHNHRiUiIiI=

The dominating eigenvalue, 1.3117, is indeed close to \316\273_\342\210\236.

We also check if the stable distribution is an eigenvector

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Yes, 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

As a second example we consider the matrix model from Allman and Rhodes, Mathematical models in biology, p 56, for a plant population.

The projection matrix for this model is

QyQ+JkkiUEc2IjYjSSZwbGFudEdGJi1JJ01hdHJpeEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYlIiIlRi83MiQiIiMhIiMiIiEkIiQ+IiEiIkY0JCIiJkYzJCIjN0YzJCIjZEY3RjRGNCQiIzlGMyQiI0BGMyQiI0tGM0Y0RjQkIiNWRjMkIiM2RjMiIiI=

The 20-generation population orbit through u0=<0,50,50,0> is

QyQ+SSJTRzYiLUknb3JiaXRQR0YlNiUmSSJQR0YlNiNJJnBsYW50R0YlLUkkPCw+R0koX3N5c2xpYkdGJTYmIiIhIiNdRjJGMSIjPyEiIg==

QyQtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSSQ8LD5HSShfc3lzbGliRzYiNiMtRiQ2JC1JJmZsb29yRzYkRiVGKTYjJiZJIlNHRio2I0kiakdGKjYjSSJpR0YqL0Y4OyIiIiIiJS9GNjtGOyIjP0Y7

The temporal graph of the population vecors

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stabilizes.

Here is a graph of the population distribution

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The stable population distribution is

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The total population through the 20 generations

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grows (approximately) exponentially: The total populations size is c* \316\273_\342\210\236^t for some constant c and some stable growth rate \316\273_\342\210\236.

Conclusion

1.The population distribution stabilizes at the stable population distribution

2. The total population grows exponentially with a stable growth rate

Let us now try to determine the stable growth rate \316\273_\342\210\236.

Since there is a stable population distribution y_\342\210\236, we expect Py_\342\210\236 to have the same population distribution. This means that we expect Py_\342\210\236 to be proportional to y_\342\210\236. The proportionality factor must be the stable growth rate \316\273_\342\210\236.

We therefore estimate the stable growth rate to be

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We choose the constant c so that the two graphs agree at t=20, so that total population size at t=20 is c* \316\273_\342\210\236^(20).

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Here is the predicted exponential growth of the total population compared to the actual total population

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We see a good match between the two curves.

Based on this, we predict that the population vector is approximately c/add(y_\342\210\236[i],i=1..4)*\316\273_\342\210\236^t*y_\342\210\236.

We compare this predicted population vector and the actual population vector

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The Fundamental Theorem of Demography says that the stable growth rate is the dominating eigenvalue and the stable distribution is its eigenvector.

We check that the stable growth rate is the dominating eigenvalue.

The eigenvalues are

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so the dominating eigenvalue is

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The dominating eigenvalue, 1.3117, is indeed close to

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We also check if the stable distribution is an eigenvector

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These two examples corroborate the Fundamental Theorem of Demography.

Here is a little program, dom, that runs the same computations as above and gives an estimate for the stable distribution and the stable growth rate. I apply it to the above two examples and to the coyote model from Allman and Rhodes, Mathematical models in biology, p 63

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