{
"cells": [
{
"cell_type": "markdown",
"id": "d3e5514e-2de3-4c33-a94b-928e9f0f6af9",
"metadata": {},
"source": [
"**Example 4.2.** The blue whale and fin whale are two similar species that inhabit\n",
"the same areas. Hence, they are thought to compete. The intrinsic growth rate\n",
"of each species is estimated at 5% per year for the blue whale and 8% per year\n",
"for the fin whale. The environmental carrying capacity (the maximum number\n",
"of whales that the environment can support) is estimated at 150,000 blues and\n",
"400,000 fins. The extent to which the whales compete is unknown. In the last\n",
"100 years intense harvesting has reduced the whale population to around 5,000\n",
"blues and 70,000 fins. Will the blue whale become extinct?"
]
},
{
"cell_type": "markdown",
"id": "f8d8e5ad-f43c-408d-a3ce-5ec2d5806ad0",
"metadata": {},
"source": [
"$B = {}$number of blue whales\\\n",
"$F = {}$number of fin whales\\\n",
"$g_B = {}$growth rate of blue whale population (per year)\\\n",
"$g_F = {}$growth rate of fin whale population (per year)\\\n",
"$c_B = {}$effect of competition on blue whales (whales per year)\\\n",
"$c_F = {}$effect of competition on fin whales (whales per year)"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "347f1328-c9d8-4b66-8bb2-5d4abd6293b6",
"metadata": {},
"outputs": [],
"source": [
"using Symbolics"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "d8036028-7f1d-4911-bf3d-7448f76a456a",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"cF (generic function with 1 method)"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"gB(B,F)=0.05*B*(1-B/150000)\n",
"gF(B,F)=0.08*F*(1-F/400000)\n",
"cB(B,F)=alpha*B*F\n",
"cF(B,F)=alpha*B*F"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "1ec28cf9-f758-4291-87c9-dbe53e4ff561",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3-element Vector{Num}:\n",
" B\n",
" F\n",
" alpha"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"@variables B,F,alpha"
]
},
{
"cell_type": "markdown",
"id": "2459bf4b-1da3-463d-9656-543632f0d3e7",
"metadata": {},
"source": [
"Model the growth and competition between whales using\n",
"a differential equation\n",
"$$\n",
" {dX\\over dt}=G(X)\n",
"$$\n",
"where\n",
"$$\n",
" X=(B,F)\n",
"\\quad\\hbox{and}\\quad\n",
" G(B,F)=\\left[\\matrix{g_B-c_B\\cr g_F-c_F}\\right]\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "e9c03c9a-fc54-4dd8-a56d-7bed5a5416a3",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"G (generic function with 1 method)"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"G(B,F)=[gB(B,F)-cB(B,F),gF(B,F)-cF(B,F)]"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "612a0f04-a190-44f1-a0e5-d8f8d3efd51f",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Vector{Num}:\n",
" \u001b[96m0.05\u001b[39mB*(\u001b[96m1\u001b[39m - \u001b[96m(\u001b[39m\u001b[96m1//150000\u001b[39m\u001b[96m)\u001b[39m*B) - B*F*alpha\n",
" \u001b[96m0.08\u001b[39mF*(\u001b[96m1\u001b[39m - \u001b[96m(\u001b[39m\u001b[96m1//400000\u001b[39m\u001b[96m)\u001b[39m*F) - B*F*alpha"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"G(B,F)"
]
},
{
"cell_type": "markdown",
"id": "d859ad89-18c1-4aca-8a27-a6ddb208fc2f",
"metadata": {},
"source": [
"Trying to solve for $G(B,F)=0$ for $B>0$ and $F>0$. Note that we\n",
"can divide the first component by $B$ and the second by $F$ to obtain\n",
"a linear system."
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "8d8a6dd0-617d-4d50-9c7e-050599a55f02",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Vector{Num}:\n",
" \u001b[96m0.05\u001b[39m - \u001b[96m3.3333333333333335e-7\u001b[39mB - F*alpha\n",
" \u001b[96m0.08\u001b[39m - \u001b[96m2.0000000000000002e-7\u001b[39mF - B*alpha"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Q=simplify.(G(B,F)./[B,F])"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "0c584b98-ae29-49b6-bc19-9e9d438a427b",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"fQ (generic function with 1 method)"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Qs=\"fQ(B,F)=\"*string(Symbolics.toexpr(Q))\n",
"eval(Meta.parse(Qs))"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "2a113263-27fa-4dfe-8519-51e69979fef1",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Vector{Num}:\n",
" \u001b[96m0.05\u001b[39m - \u001b[96m3.3333333333333335e-7\u001b[39mB - F*alpha\n",
" \u001b[96m0.08\u001b[39m - \u001b[96m2.0000000000000002e-7\u001b[39mF - B*alpha"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"fQ(B,F)"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "f42006f4-c9b5-47f5-8016-3bf9f2889c2d",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Vector{Num}:\n",
" 0.05\n",
" 0.08"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"b=fQ(0,0)"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "314e3836-5fa3-46d1-a855-05fc76b0d4d6",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2×2 Matrix{Num}:\n",
" -3.33333e-7 -alpha\n",
" -alpha -2.0e-7"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A=Symbolics.jacobian(fQ(B,F),[B,F])"
]
},
{
"cell_type": "markdown",
"id": "b948a81c-96a7-4a12-8299-057c7b28fa12",
"metadata": {},
"source": [
"Note that $Q(X)=Q(0)+(DQ(0))(X-0)$ since $Q$ is linear."
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "83ae0307-bb7d-4aea-8e93-d54fc96a2e20",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Vector{Num}:\n",
" (\u001b[96m0.05\u001b[39m + (-(\u001b[96m0.08\u001b[39m - \u001b[96m150000.0\u001b[39malpha)*alpha) / (\u001b[96m2.0000000000000002e-7\u001b[39m - \u001b[96m3.0e6\u001b[39m(alpha^\u001b[96m2\u001b[39m))) / \u001b[96m3.3333333333333335e-7\u001b[39m\n",
" (\u001b[96m0.08\u001b[39m - \u001b[96m150000.0\u001b[39malpha) / (\u001b[96m2.0000000000000002e-7\u001b[39m - \u001b[96m3.0e6\u001b[39m(alpha^\u001b[96m2\u001b[39m))"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"X0=-A\\b"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "2a00e7db-fd61-4fea-ad12-be37389b1136",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"fX0 (generic function with 1 method)"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"X0s=\"fX0(alpha)=\"*string(Symbolics.toexpr(X0))\n",
"eval(Meta.parse(X0s))"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "37d454fe-2773-4a9b-b933-dc6c680e3243",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Vector{Num}:\n",
" (\u001b[96m0.05\u001b[39m + ((-\u001b[96m0.08\u001b[39m + \u001b[96m150000.0\u001b[39malpha)*alpha) / (\u001b[96m2.0000000000000002e-7\u001b[39m - \u001b[96m3.0e6\u001b[39m(alpha^\u001b[96m2\u001b[39m))) / \u001b[96m3.3333333333333335e-7\u001b[39m\n",
" (\u001b[96m0.08\u001b[39m - \u001b[96m150000.0\u001b[39malpha) / (\u001b[96m2.0000000000000002e-7\u001b[39m - \u001b[96m3.0e6\u001b[39m(alpha^\u001b[96m2\u001b[39m))"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"fX0(alpha)"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "19d63199-2d8e-4441-9a46-6a28e6ef2b92",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Vector{Float64}:\n",
" 35294.11764705884\n",
" 382352.94117647054"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"fX0(1e-7)"
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "127c0c8d-4783-4b56-b11a-9eece4c0b0dc",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Vector{Float64}:\n",
" 150000.0\n",
" 400000.0"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"fX0(0.0)"
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "69185a68-6184-44d1-8b04-4146559aad97",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Vector{Float64}:\n",
" 138207.31096644967\n",
" 393089.6344516775"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"fX0(1e-8)"
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "bfca171f-bc7f-4f6f-98a4-716db1333a00",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Vector{Float64}:\n",
" 75000.00000000001\n",
" 25000.0"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"fX0(1e-6)"
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "12cbf4f1-d99b-486a-a97e-ad02570d2541",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Vector{Float64}:\n",
" 7905.270180120063\n",
" 4736.490993995997"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"fX0(1e-5)"
]
},
{
"cell_type": "code",
"execution_count": 19,
"id": "fe0bdbe6-8bd3-4527-89fb-41f3afa3c2d2",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Vector{Float64}:\n",
" 799.0053267022054\n",
" 497.33664891099266"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"fX0(1e-4)"
]
},
{
"cell_type": "code",
"execution_count": 20,
"id": "aee5c4f2-e656-4ddc-9d47-95f25230f8cb",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1.25e-7"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"0.05/400000"
]
},
{
"cell_type": "code",
"execution_count": 21,
"id": "7f1a80ae-4dd6-48ff-b805-d75aefb31a26",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
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\")
"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using Plots\n",
"plot(alpha->fX0(alpha)[1],alpha->fX0(alpha)[2],1e-8:0.1e-08:1e-7)\n",
"scatter!((fX0(1e-8)[1],fX0(1e-8)[2]))"
]
},
{
"cell_type": "markdown",
"id": "cd762afe-6460-459a-8418-a2036c035d32",
"metadata": {},
"source": [
"Now lets set $\\alpha=10^{-7}$ and linearize $G$ about the\n",
"corresponding equilibrium to see stability."
]
},
{
"cell_type": "code",
"execution_count": 22,
"id": "67b3f855-e571-404a-9af9-2e9a65f60c19",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Vector{Num}:\n",
" \u001b[96m0.05\u001b[39mB*(\u001b[96m1\u001b[39m - \u001b[96m(\u001b[39m\u001b[96m1//150000\u001b[39m\u001b[96m)\u001b[39m*B) - \u001b[96m1.0e-7\u001b[39mB*F\n",
" -\u001b[96m1.0e-7\u001b[39mB*F + \u001b[96m0.08\u001b[39mF*(\u001b[96m1\u001b[39m - \u001b[96m(\u001b[39m\u001b[96m1//400000\u001b[39m\u001b[96m)\u001b[39m*F)"
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"alpha=1e-7\n",
"G(B,F)"
]
},
{
"cell_type": "markdown",
"id": "8929c63a-f1ae-4926-833d-835e9cd24e18",
"metadata": {},
"source": [
"Now linearlize $G$ as\n",
"$$\n",
" G(X)\\approx G(X_0)+(DG(X_0))(X-X_0)\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 23,
"id": "534e10b2-b599-4726-8f9d-27f04320fd4c",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Vector{Float64}:\n",
" 35294.11764705884\n",
" 382352.94117647054"
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"myX0=fX0(alpha)"
]
},
{
"cell_type": "code",
"execution_count": 24,
"id": "1ca0e598-00e0-4882-ad0f-ee38517685c9",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2×2 Matrix{Num}:\n",
" -\u001b[96m3.33333e-7\u001b[39mB + \u001b[96m0.05\u001b[39m(\u001b[96m1\u001b[39m - \u001b[96m(\u001b[39m\u001b[96m1//150000\u001b[39m\u001b[96m)\u001b[39m*B) - \u001b[96m1.0e-7\u001b[39mF … \u001b[96m-1.0e-7\u001b[39mB\n",
" \u001b[96m-1.0e-7\u001b[39mF -\u001b[96m1.0e-7\u001b[39mB - \u001b[96m2.0e-7\u001b[39mF + \u001b[96m0.08\u001b[39m(\u001b[96m1\u001b[39m - \u001b[96m(\u001b[39m\u001b[96m1//400000\u001b[39m\u001b[96m)\u001b[39m*F)"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"DG=Symbolics.jacobian(G(B,F),[B,F])"
]
},
{
"cell_type": "code",
"execution_count": 25,
"id": "bf5ac2f5-9252-40ec-9424-1feb6662cd80",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2×2 Matrix{Float64}:\n",
" -0.0117647 -0.00352941\n",
" -0.0382353 -0.0764706"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"DG0=substitute(DG,[B=>myX0[1],F=>myX0[2]])\n",
"myDG0=eval(Symbolics.toexpr(DG0))"
]
},
{
"cell_type": "code",
"execution_count": 26,
"id": "2e12b985-9669-45f7-890e-89d1262944be",
"metadata": {},
"outputs": [],
"source": [
"using LinearAlgebra"
]
},
{
"cell_type": "code",
"execution_count": 27,
"id": "2351a214-4f93-4e29-b45a-4a4bc78a85fb",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Vector{Float64}:\n",
" -0.07849294196161641\n",
" -0.009742352156030629"
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"eigvals(myDG0)"
]
},
{
"cell_type": "markdown",
"id": "55b628cc-0feb-4079-b77d-31e622afcf49",
"metadata": {},
"source": [
"This means along the eigendirections the populations\n",
"decrease towards the fixed point $X_0$. In order words,\n",
"the fixed point is stable in a neighborhood."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "8f2f2670-6064-4a73-9db8-3229c7fc142f",
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"id": "3c897e59-223b-487a-90a6-fe18d753698a",
"metadata": {},
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Julia 1.10",
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