Rabbits, Foxes and Mathematical Modeling

Peter PangUniversity Scholars Programme and

Department of Mathematics, NUS

SMS Workshop

July 24, 2004

Mathematical Modeling of Population Dynamics

How does population grow?

Denote the population by “x”. Suppose the population x(t) at time t changes to x + Δx in the time interval [t, t + Δt]. Then the growth rate is

ttx

x

)(

Time Population Growth Rate 0.11 102 113 12.14 13.315 14.6416 16.1057 17.7168 19.4879 21.436

10 23.57911 25.93712 28.53113 31.38414 34.52315 37.97516 41.77217 45.9518 50.54519 55.59920 61.15921 67.27522 74.00223 81.40324 89.54325 98.49726 108.3527 119.1828 131.1

0

200

400

600

800

1000

1200

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

The growth rate depends on many things, such as Per capita food supply – call it “s” A minimum supply of food, say s0, is needed to sustain life Say growth rate is proportional to s – s0

Call the constant “a” the growth coefficient

)()( 0ssattx

x

)()( 0 txssat

x

Time Population Food Supply 81 10 Min Food Supply 52 11 Growth Coeff 0.03333 12.14 13.315 14.646 16.1047 17.7158 19.4869 21.434

10 23.57811 25.93512 28.52813 31.38114 34.51915 37.9716 41.76717 45.94318 50.53719 55.5920 61.14921 67.26322 73.98823 81.38624 89.524

Population

0

200

400

600

800

1000

1200

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

Is infinite growth realistic? Suppose population reaches saturation at x0

Say the growth coefficient is proportional to x0 – x

We can interpret the x2 term as a number proportional to the average number of encounters between x individuals. Hence it measures a kind of social friction.

2000

00

)()()()(

)()))(((

txssbtxssbx

txsstxxbt

x

Time Population Food Supply 81 10 Min Food Supply 52 11 Pop Saturation 10003 12.099 Coeff 3.37E-054 13.3065 14.6326 16.0897 17.6888 19.4439 21.369

10 23.48111 25.79712 28.33513 31.11714 34.16215 37.49516 41.1417 45.12518 49.47719 54.22720 59.40821 65.05222 71.19523 77.87524 85.129

0

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1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

Foxes and Rabbits

Let’s look at the fox population Let’s assume that the fox population doesn’t get really huge so that

the issue of “population saturation” can be ignored Recall that the model for unlimited growth is

Now, suppose that the only food for foxes is rabbits; then s is proportional to the population of rabbits

Denote rabbit population by “y”

)()( 0 txssat

x

xdcy

tdxtytcx

txstbyat

x

)(

)()()(

)())(( 0

As for the rabbit population, let’s again assume we have unlimited growth while the rabbits are being eaten by the foxes -- we further assume that the number of rabbits eaten is proportional to the fox population

ygxf

tytxgtyft

y

)(

)()()(

The Lotka-Volterra (Predator-Prey) Equations

Population of foxes – x Population of rabbits – y

where c, d, f, g are constant parameters

ygxft

y)(

xdcyt

x)(

Time Fox Rabbit c 5E-061 35000 70000 d 0.32 36750 80500 f 0.53 40517 91166 g 1E-054 46831 998125 56153 1029756 68219 966397 80716 790338 88397 547579 86080 33732

10 74774 2156111 60403 1622012 47181 1453213 36455 1494214 28242 1696615 22165 2065716 17805 2640717 14814 3490918 12956 4719219 12126 6467420 12410 8916921 14220 12268822 18676 16658623 28630 21876724 51357 265518

0

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1 3 5 7 9 11 13 15 17 19

Fox

Rabbit

Let’s introduce a saturation for the rabbit population:

Time Fox Rabbit c 0.0000051 50000 60000 d 0.32 50000 60000 f' 0.00001253 50000 60000 g 0.000014 50000 60000 rab-sat 1000005 50000 600006 50000 600007 50000 600008 50000 600009 50000 60000

10 50000 6000011 50000 6000012 50000 6000013 50000 6000014 50000 6000015 50000 6000016 50000 6000017 50000 6000018 50000 6000019 50000 6000020 50000 6000021 50000 6000022 50000 6000023 50000 6000024 50000 60000

44000

46000

48000

50000

52000

54000

56000

58000

60000

62000

1 3 5 7 9 11 13 15 17 19Fox

Rabbit

Functional Response

Lotka-Volterra System (with unlimited growth for prey)

The term p(y) measures the number of prey susceptible to each predator as the prey population changes -- this is known as the functional response

p(y) = gy says that the number of prey for each predator is a constant proportion of the prey population

What if the prey population really shoots up?

xdxyct

x

xygyft

y

yp )(

Michaelis-Menten or Holling type II functional response

yh

myyp

)(

y p(y) m 100 0 h 1

0.5 3.3333331 5

1.5 62 6.666667

2.5 7.1428573 7.5

3.5 7.7777784 8

4.5 8.1818185 8.333333

5.5 8.4615386 8.571429

6.5 8.6666677 8.75

7.5 8.8235298 8.888889

8.5 8.9473689 9

9.5 9.04761910 9.090909

10.5 9.13043511 9.166667

11.5 9.2

0

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8

9

10

Sigmoidal functional response

2

2

)(yjyh

myyp

y p(y) m 100 0 h 1

0.5 0.666667 j 51 1.428571

1.5 2.0930232 2.666667

2.5 3.1645573 3.6

3.5 3.983744 4.324324

4.5 4.6285715 4.901961

5.5 5.1489366 5.373134

6.5 5.5775587 5.764706

7.5 5.9366758 6.095238

8.5 6.2419019 6.377953

9.5 6.50450510 6.622517

10.5 6.73282411 6.836158

11.5 6.933159

0

1

2

3

4

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6

7

8

9

Holling type III functional response

2

2

)(yh

myyp

y p(y) m 100 0 h 1

0.5 21 5

1.5 6.9230772 8

2.5 8.620693 9

3.5 9.2452834 9.411765

4.5 9.5294125 9.615385

5.5 9.686 9.72973

6.5 9.7687867 9.8

7.5 9.8253288 9.846154

8.5 9.8634819 9.878049

9.5 9.89041110 9.90099

10.5 9.91011211 9.918033

11.5 9.924953

0

1

2

3

4

5

6

7

8

9

10

Holling type IV or Monod-Haldane type functional response

2)(

yjyh

myyp

y p(y) m 100 0 h 1

0.5 4 j 01 5

1.5 4.6153852 4

2.5 3.4482763 3

3.5 2.6415094 2.352941

4.5 2.1176475 1.923077

5.5 1.766 1.621622

6.5 1.502897 1.4

7.5 1.3100448 1.230769

8.5 1.160419 1.097561

9.5 1.04109610 0.990099

10.5 0.9438211 0.901639

11.5 0.863039

0

1

2

3

4

5

6

7

8

9

10

t x y m 100 g 0.0000001 sat 2000001 50000 60000 h 1 b 0.012 49525 60756.667 j 0 c 0.33 49054.2 61521.1494 48587.58 62293.3515 48125.11 63073.1746 47666.75 63860.5157 47212.47 64655.2668 46762.25 65457.329 46316.06 66266.561

10 45873.87 67082.87311 45435.65 67906.13512 45001.36 68736.22413 44570.99 69573.01214 44144.5 70416.36815 43721.86 71266.15816 43303.05 72122.24517 42888.03 72984.48718 42476.78 73852.7419 42069.26 74726.85620 41665.46 75606.68621 41265.34 76492.07422 40868.87 77382.86523 40476.03 78278.89724 40086.78 79180.009

0

20000

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1 5 9 13 17 21 25 29 33 37 41 45 49

x

y

Ratio-Dependent Theory

Recall that functional response measures the number of prey susceptible to each predator as the prey population changes.

We have seen various types of functional response functions of the type p(y).

The ratio-dependent theory asserts that functional response should be dependent of the ratio of prey to predator (especially if the predator needs to search for the prey), i.e., instead of being just a function of y, p should be a function of y/x :

x

ypp

t x y m 0.1 g 0.0000008 sat 2000001 35000 70000 h 1 b 0.012 35350 74946.667 j 1 c 0.33 35717.11 80042.4964 36100.84 85254.1795 36500.68 90544.0766 36916.09 95871.1617 37346.52 101192.228 37791.42 106463.199 38250.23 111640.68

10 38722.41 116683.3511 39207.4 121553.2912 39704.69 126217.1313 40213.74 130646.9114 40734.08 134820.6215 41265.21 138722.3916 41806.69 142342.4117 42358.09 145676.5618 42918.99 148725.8619 43489.02 151495.7620 44067.81 153995.3821 44655.02 156236.6822 45250.34 158233.7423 45853.47 160002.0424 46464.13 161557.86

02000040000

6000080000

100000120000

140000160000180000

1 12 23 34 45 56 67 78 89 100

x

y

Spatial Dependence

What can happen to a spatially inhomogeneous population? Diffusion

Cross Diffusion Partial differential equations

Two Predators and One Prey

Defense switching Cross diffusion