An exactly solvable model of population dynamics with density-dependent

migrations and the Allee effect

Sergei Petrovskii, Bai-Lian Li

Kristina Little

A Basic Introduction to the Paper

o Single species model of population dynamics

o Taking into account two types of migration

o Advection, caused by environmental factors

o Migration, associated with a biological mechanism

o Also taking into consideration the Allee Effect

o So the result is an advection-diffusion-reaction system of PDEs

Quick Recap of the Allee Effect

Refers to the positive relationship between population density and the reproduction and survival of individuals. An “Allee” or underpopulation effect arises when the per capita birth rate (B) increases with population density to some maximum value, the reproductive potential of the species, and the death rate (D) remains constant. The point U where the two rates intersect is an unstable equilibrium, separating two different kinds of dynamic behavior, growth versus extinction.

Deriving the Main Equations

U(R, t) = population density, dependent on position, R = (X, Y, Z), and time, T.

where J is the population density flux and f(U) is the per capita birth rate.

If the flux is random, it will take the form

But What if the Flux is not Random?

Motion of individuals can’t always be considered random. Advection/Migration is observed when individuals exhibit a correlated motion toward a certain direction.

For simplicity, assume that at a given position all individuals move with the same speed, A. Then the advection/migration flux is given by

And total flux (random and nonrandom) is given by

Deriving the Main Equations (cont’d.)

So now we can model the dynamics of a given population with the following equation (simplifying R to one dimension):

Where A is positive when advection/migration is going in the positive x-direction.

This equation describes the propagation of traveling population fronts and may correspond either to a species invasion or a species retreat.

But…

… what about the case when a population is involved in both types of motion (invasion and retreat)?

For example, can advection (in the form of a strong wind) keep a species from spreading when invasion would otherwise take place?

Deriving the Main Equations (cont’d.)

Two more assumptions:

1) the growth rate is damped by the Allee effect, so

where K is the species carrying capacity, U0 is the measure of the Allee effect, and α is a constant.

2) the speed of migration is given by the following equation:

where A0 is the speed of advection and A1U is the speed of migration due to biological mechanisms (so A1 is the per capita migration speed).

Deriving the Main Equations (cont’d.)

Putting everything together we have:

which becomes

after the following substitutions of dimensionless parameters

The No-Migration Case

(i.e., the easy case)

When there is no migration (i.e., all motion is random), a0=a1=0 and our equation reduces to

Which has an exact solution of

where

The No-Migration Case (cont’d.)

Obviously, u(x,t) must be non-negative for any x and t. So our

solution is only valid for β ≥ 0.

Graph 1: Population density vs. space for different t.

1: t=0; 2: t=40; 3: t=80; 4: t=120; 5: t=160; 6: t=700

β= 0.2, φ1=100, φ2=-100

The No-Migration Case (cont’d.)

Since λ1< λ2, in the large-time limit our equation reduces to

which describes a traveling population front moving with speed n2 given by

(a) species invasion

(b) species retreat

The Migration Case: Interplay between diffusion and migration

(i.e., the hard case)

Starting first with density-independent advection, we have

where a0 is the speed of advection. Considering traveling wave coordinates (x, t) (z, t) where z=x-a0t, so that u=û(z, t), we have

The Density-Independent Migration Case (cont’d.)

This is exactly the equation we solved earlier, so…

where n2+a0 is the speed of the front (n2<-a0 corresponds to species invasion and n2>-a0 corresponds to species retreat). This is equivalent to β<½(1-√2a0)

What does this mean?: Simply, that the weaker the Allee effect is for a given population, the higher its capability to invade new area.

The Density-Dependent Migration Case

Now taking into account density-dependent migration (and no advection), we have

To solve this equation, let’s introduce a new variable p(x, t)

Substituting gives the following system:

The Density-Dependent Migration Case (cont’d.)

Wlog, we choose plus and therefore we can subtract the first equation from the second, yielding

The first equation has a solution of the form

Substituting this into the second equation yields

The Density-Dependent Migration Case (cont’d.)

Considering the previous equations along with the fact that coefficients B0+σ, B1, and B2 must have the same sign (to ensure positiveness), the exact solution of our original equation,

is

where ψi=x-qit+εi, qi=(1+β)v-(3+a1v)wi, i=1,2, and ε1,2 are arbitrary constants.

The Density-Dependent Migration Case (cont’d.)

The structure of this solution is similar to the solution of the No-Migration Case, and has similar properties. In the large-time limit solution, the equation describes a single traveling population front:

propagating with speed q2. The population invades a region of low population density when q2<0,

The Migration Case- Recap

In the Density-Independent Migration Case, β could be positive or negative depending on a0. But in this case, since β is always positive, the interplay between advection and migration is different. Even a strong density-dependent migration moving to a region with high population density cannot block the species invasion caused by the random dispersion of individuals when the Allee effect is sufficiently small.

Can We Bring This All Together?

In a general case, migrations can take place due to both density-dependent and density-independent factors. In terms of our original equation

this corresponds to a0≠0 and a1≠0. Our earlier results lead to the exact solution:

where our notation is the same used in the case of density-dependent migration.

In the Large-Time Limit

In the large-time limit, the solution again describes the propagation of a population front

and the condition of a successful invasion is q2<-a0, or

Obviously, r ≥ 0 for v ≤ 1/a0 and r is negative otherwise. But since β is assumed to be non-negative, a0 cannot be arbitrarily large for any fixed β and v.

What Does this Mean to the Populations?

Since for any β and v, sufficiently large a0 violates the inequality, the species invasion can always be blocked or reversed in case of sufficiently strong counteractive advection, provided that the density-dependent migrations are either absent or enhance the species retreat.

On the other hand, for any fixed positive value of a0, the inequality will hold for sufficiently small v (small v large negative a1). This corresponds to the case when the density dependent migration take place towards the region where the species is absent. So even strong counteractive advection cannot stave off the spread of a given population in the case of cooperative impact of biological factors enhancing the species invasion.

Conclusions

o Species invasion caused solely by isotropic random motion of the individuals can be blocked by sufficiently strong counteracting advection.

o When invasion caused by random isotropic motion is supported by a correlated motion by the individuals towards the region of low population density, the species spread cannot be blocked by purely environmental factors (provided per capita migration speed is large enough).

o Density-dependent migration alone cannot block diffusive spreading in the case that the Allee effect is “not too strong.”

CONGRATULATIONS!You Made It!!!!

Any Questions?