Competencia y Predacion

8/17/2019 Competencia y Predacion

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2/155

r = r(t, x)


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n


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7/155

dx

dt = rx

x

t

r

r > 0

r 0

x0 > 0

x(t) = 0

r < 0

x(t) = 0

x(t)

x

t

x0

Crecimiento exponencial, r

0

2 4 6 8 10t

2

4

6

8

x

Decrecimiento exponencial, r

0

2 4 6 8 10t

2

4

6

8

x


8/155

r

f (x)

x

x(t)

dx

dt = f (x) x

dx

dt

= r 1 − x

K x

r > 0

K > 0

x(t)

x(t) = K

K x0

− 1

e−rt + 1, x0 = 0

x(t)

x = 0

x = K

x K

x 0

Comportamiento decreciente de las soluciones

Comportamiento creciente de las soluciones

2 4 6 8t

0.5

1.0

1.5

2.0

x


9/155

x(t) = 0

x(t) = K

x(t)

x0 2.0

x0 10.0

x0 0.02

2 4 6 8t

0.5

1.0

1.5

x K

x

x(t)

x

dx

dt = −r

1 − x

T

1 − x

K

x

T > 0

K

K > T

x |x − K |T

K−

T |x − T |− K

K−

T = x0 |x0 − K |T

K−

T |x0 − T |− K

K−

T e−rt


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x0

• x0 < T

• T < x0 < K

• x0 > K

x = 0

x = T

x = K

x Π

x 0

x K


Comportamiento creciente de las soluciones


2 4 6 8t

0.5

1.0

1.5

2.0

2.5

x

x = 0

x = K

x = T

T = 1

K = 2

x(t)

xt K

xt T

1 2 3 4 5 6 7t

0.5

1.0

1.5

2.0

2.5

3.0

x


11/155

h(t)

h

x(t)

dx

dt = r

1 − x

K

x − h(t)

h(t) = H

H

dx

dt = r 1 − xK x − H

x1 =K −

K 2 − 4HK

r

2 ; x2 =

K +

K 2 − 4HK r

2

K 2 − 4HK

r rK 4

x0 r = 1

K = 1 H = 0,3 > 14

2 4 6 8 10 12t

0.5

1.0

1.5

x

H > rK 4

H = rK 4 x1 =

K 2 x0 >

K 2

dxdt


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xt x1 K

2

2 4 6 8 10 12

t

0.2

0.4

0.6

0.8

1.0

1.2

1.4

x

H = rK 4

H < rK 4 x1 x2

• x0 < x1 dx

dt 0

x2

• x0 > x2 dx

dt

x2

x0 r = 1

K = 1 H = 0,2

x1 = 0,276 x2 = 0,724

x 2

x 1

2 4 6 8 10 12

t

0.2

0.4

0.6

0.8

1.0

1.2

x

H < rK 4

x1

x2


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x0 < x1

x0 H c = rK 4

H

H H c

x∞ H > H c

H H c

H H c

H H c

2 4 6 8 10 12

t

0.5

1.0

1.5

2.0

2.5

x

H > H c

h(t)

h = h(t, x)

h(t) = H

∆y = yn+1 − yn = r 1 − ynK yn


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yn n

r

K (1 + r)yn = xn;

r

K (1 + r)yn+1 = xn+1

xn+1 = (1 + r)xn(1 − xn)

1 + r = R

xn+1 = Rxn(1 − xn)

R

x −→ f (x) = Rx(1 − x)

R

x

x ∈ [0, 1]

x

x < 0

1 − x > 0 Rx(1 − x)

xn+1 = Rxn(1 − xn)

[0, 1]

x > 1

1−x 4 x

[0, 1]

R

f (x)

x

xn f (xn)

f (xn)

xn+1

xn n

• 0 R 1 x −→ Rx(1 − x) x = 0

R = 0,5

x

[0, 1]


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R 0.5

1 2 3 4 5 6n

0.1

0.2

0.3

0.4

0.5

0.6

xn

• 1 < R 0

xn+1 = Rε(1 − ε)

ε

R(1 − ε) > 1

R

xn+1 > ε

xn x0

xf = 1

−1R

f (x) =R(1− 2x) f (xf ) = 2 −R |f (xf )|


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R 2.8

5 10 15 20n

0.2

0.4

0.6

0.8

1.0

xn

xf = 1 − 1R

R = 2,8

R > 3

x

• R = 3

R

x = 1 − 1R

x1 x2

R = 3,1

R 3.1

10 20 30 40 50n

0.2

0.4

0.6

0.8

1.0

xn

• R

x1 x2

R = 3,44

R = 3,5


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x0

R 3.5

10 20 30 40 50 60n

0.2

0.4

0.6

0.8

1.0

xn

•

R = 3,54

R = 3,564

R 3.564

100 200 300 400 500 600n

0.2

0.4

0.6

0.8

1.0

xn

• R

R = 3,5699457 · · ·

R = 3,6786

R = 1 +√ 8 =


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3,8284 · · · R = 3,62

R = 4

R 3.62

500 1000 1500 2000 n

0.2

0.4

0.6

0.8

1.0

xn

R 4

500 1000 1500 2000 n

0.2

0.4

0.6

0.8

1.0

xn


19/155

dxdt

r

r = r(t)

r(t)

x(t)

dx

dt = r(t) x

r(t)

r(t) = n(t) − m(t)

• n(t)

• m(t)

• r(t)

x(0) = x0

x(t) = x0e´ t0 r(s)ds

r(t) r0 > 0 t > t0

x(t) = x0e´ t0 r(s)ds x0e

´ t0 r0ds = x0e

r0t −→ ∞

t → ∞

r(t) = log(t + 3)

t > 0

1 2 3 4t

0.5

1.0

1.5

x


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r(t) −r1 t0

0 x(t) = x0e´ t0 r(s)ds

x0e´ t0 −

r1ds

= x0e−r1t

−→ 0 t → ∞

r(t) = −0,5log(t + 3)

1 2 3 4t

0.5

1.0

1.5

x

r(t) → 0 ´ ∞0 |r(s)|ds < ∞

x(t)

a =´ ∞0 |r(s)|ds

ĺımt→∞

x(t) = x0e´ ∞

0 r(s)ds x0e

´ ∞

0 |r(s)|ds = x0ea < ∞

r(t)

r(t)

T

r(t + T ) = r(t)

R =

ˆ T 0

r(s)ds

x(t) → ∞ R > 0 k

[0, t)

[0, t) = [0, kT ) ∪ [kT,t)

x(t) = x0e´ t0 r(s)ds = x0e

´ kT 0

r(s)dse´ tkT

r(s)ds = x0ekRe

´ tkT

r(s)ds

r(t)

´ tkT

r(s)ds =´ t−kT 0 r(s)ds

t1 = t − kT t1 ∈ [0, T )

x(t) = ekR

x0e´ t10 r(s)ds

= ekR

x(t1)


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∞ t → ∞ k → ∞ x(t1) > 0 t1

x(t) → 0 R < 0

R = 0

x(t)

2π

r(t) =

(sen t)2,5

t ∈ [0, π)

−(sen t)2,5 t ∈ [π, 2π)

R =

´ 2π0 r(s)ds = 0

2 4 6 8 10 12t

1

2

3

4

5

6

x

r = r(t, x)

r

x

n(t, x)

m(t, x)

x

∂n(t, x)

∂x 0

r(t, x) = n(t, x) − m(t, x)

∂r(t, x)

∂x


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x

1

px

d( px)

dt =

1

x

dx

dt

p > 0

r(t, x) = r(t,px)

xmax

r(t, x) = r

t,

x

xmax

; xmax > 0

r(t, x)

x

r(t, x) = r

t, x

xmax

x = 0

r(t, x) = r0(t) + r1(t)

x

xmax

+ r2(t)

x

xmax

2+ · · ·

r0(t) > 0 x

(t, x)

r(t, x)

x

xxmax

1

r(t, x) = r0(t) + r1(t)

x

xmax

= r0(t)

1 +

r1(t)

r0(t) xmaxx

∂r(t,x)∂x


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T

dxdt = r(t)1 − xK x

x(t) = K

K x0

− 1

e−´ t0 r(s)ds + 1

r(t)

t = nT +t1 t1 ∈ [0, T ) r(t) = r(t1)

R =

´ T 0 r(s)ds

´ t0 r(s)ds = nR +

´ t10 r(s)ds

x(t) = x(nR + t1) = K

e−nR

K x0

− 1

e−´ t10

r(s)ds + 1

• R > 0 x(t) → K t → ∞•

R 0


24/155

dxdt = r1 − xK (t) x; K (t) > 0

y = x−1

dy

dt = −ry + r

K (t)

y(t)

µ(t) = ert

y

y(t) = e−rt

r

ˆ t0

ers

K (s)ds + y0

x

x(t) = x0e

rt

x0r´ t0

ers

K (s)ds + 1

x(t)

K (t)

T

t = nT + t1 t1 ∈ [0, T )

ˆ t0

ers

K (s)ds =

n

j=1ˆ jT ( j

−1)T

ers

K (s)ds

I 1

+

ˆ nT +t1nT

ers

K (s)ds

I 2

I 1 v = s − ( j − 1)T

I 1 =

ˆ jT ( j−1)T

ers

K (s)ds =

ˆ T 0

er(v+( j−1)T )

K (v + ( j − 1)T ) dv = er( j−1)T

ˆ T 0

erv

K (v)dv

K (t)

T

K (v + ( j − 1)T ) = K (v)

´ T 0

erv

K (v)dv = pr

p

I 1 = p

rer( j−1)T

n j=1

p

rer( j−1)T =

p

r

n j=1

erT

j−1=

p

r

1 − ernT 1 − erT

I 2 v = s − nT

I 2 =

ˆ nT +t1nT

ers

K (s)ds =

ˆ t10

er(v+nT )

K (v + nT )dv = ernT

ˆ t10

erv

K (v)dv

p

r

1 − ernT 1 − erT

+ ernT ˆ t1

0

erv

K (v)dv


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x

L

sup

dxdt

|x(a) − x(b)| L |a − b|

|x(nkT + sk) − x(nkT + s∗)| L

|nkT + sk

−nkT

−s∗

|= L

|sk

−s∗

| →0

sk

→s∗

sk

s∗

sk

s∗

x∞(t) = x∞(nT + t1) = x∞(t1) = ert1

perT −1 + r

´ t10

erv

K (v)dv

x∞(t) T

x0

K =´ T 0 K (t)dt K (t) T

r = 0,5

K (t) =

2+s en2t

dx

dt = 0,5

1 − x

2 + sen 2t

x

x(0) = 1,0

x(0) = 2,2

x

t

K t

2 4 6 8 10t

0.5

1.0

1.5

2.0

2.5

3.0

x


27/155

x∞(t)

x∞(t)

x∞ = 1T

ˆ T 0

x∞(t)dt ≈ 1,747

K (t)

K (t)

x(t)


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•

•

•

dx

dt = f (x)

x ∈ Rn f : Rn → Rn f


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t → x(t)

I ⊂ R

Rn

dx(t)dt

= f (x(t))

x(t)

t = 0

x(0) = x0 x0 ∈ Rn

G ⊂ Rn

f : G → Rn

x ∈ G x0 ∈ G

ẋ = f (x)

x(t0) = x0 x : I → U

I

t0

f

x1 f (x1) = 0

x(t) = x1

Rn

x(t)

n = 2

{t, x(t)} ∈ Rn+1

z

dxdt

= f (x)

U

W

W

U

x0 ∈ W x(t) ∈ U t 0

z

W

z

x(t) → z t → ∞ x0 ∈ W

x0 x(t) → z z

x0 W

z

x0 U y ∈ U

y(t)

y(0) = y0 w ∈ W

x(t)

y0 t0

y(t) = x(t + t0) y(0) = y0 y(t) → z

t → ∞

x(t) → z

t → ∞


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z

x

y

dxdt

= x(a − by)dydt

= y(−c + dx)

a

b

c

d

dx

dt = ẋ

• y = 0 dxdt =ax,

a

x

• x = 0 dydt

=

−cy

c

y

• b > 0 −bxy

xy

• d > 0 dxy

dxdt

dydt

x y 1x dxdt 1y dydt

1x

dxdt

= a − by1y

dydt

= −c + dx

x y

x

y

Competencia y Predacion

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