Competition for Territory: The Levins Model for Two Species

colonization strategies

p

1

⥯

p

2

p

1

⟶

p

2

SubsuperscriptBox[SubscriptBox[\(p\), \(\(1\)\( \)\)], \(⟵\), \(⟶\)]

p

2

species 1 rates

colonization

0.7

extinction

0.5

species 2 rates

colonization

0.7

extinction

0.1

relative overcolonization

T

0.6

initial conditions

reset locators

equilibria:

{0.,0.857143}

{0.285714,1.58603×

-17

10

}

{0.,0.}

This population dynamics model (being the extension of the well-known Levins metapopulation model) describes the competition for territory of two species having different colonization strategies. The model equations are:

p

1

'(t)=

c

1

p

1

(t)(1-

p

1

(t)-

p

2

(t))-

e

1

p

1

(t)+o

p

1

(t)

p

2

(t)

,

p

2

'(t)=

c

2

p

2

(t)(1-

p

1

(t)-

p

2

(t))-

e

2

p

2

(t)-o

p

1

(t)

p

2

(t)

,

where

p

1

(t)

and

p

2

(t)

are the portion of the territory at time

t

occupied by species 1 and 2, respectively. The occupation is exclusive. Hence, the assumptions

0≤

p

1

(t),0≤

p

2

(t)

and

p

1

(t)+

p

2

(t)≤1

hold.

The meaning of the parameters is as follows:

c

1

,

c

2

≥0

, the global rates of colonization, describe how the species can colonize empty patches. The global rates of extinction are

e

1

,

e

2

≥0

. The global rate of relative overcolonization

o

describes how the species can occupy the patches occupied by the other species.