The terms

r_{i} N_{i}

give intrinsic per-capita population growth. The terms

- r_{i} N_{i}^{2} / K_{i}

model intraspecific crowding in a logistic equation, while

- r_{i} b_{ij} N_{i} N_{j} / K_{i}

K_{i}

b_{ij}

measures the competitive effect of one unit of species

j

relative to one unit of species

i

in that species' crowding term. The last term

- δ N_{2}

is per-capita removal of grey squirrels: constant effort, rather than a constant number removed per unit time.

The intrinsic rates

r_{1}, r_{2}

and removal rate

δ

have units of inverse time. The capacities

K_{1}, K_{2}

have the same density units as the respective

N_{i}

; with both densities expressed in the same units,

b_{12}, b_{21}

are dimensionless. The two equations form a Competitive Lotka-Volterra model with density regulation and harvesting.

ii

Solution

Since

d N_{i} / d t = K_{i} r_{1} d u_{i} / d τ

K_{i} r_{1}

gives

\frac{d u _{1}}{d τ} = u_{1} (1 - u_{1} - \frac{b _{12} K _{2}}{K _{1}} u_{2}), \frac{d u _{2}}{d τ} = \frac{r _{2}}{r _{1}} u_{2} (1 - \frac{δ}{r _{2}} - u_{2} - \frac{b _{21} K _{1}}{K _{2}} u_{1}) .

Thus the dimensionless parameters are

α_{12} = b_{12} K_{2} / K_{1}, α_{21} = b_{21} K_{1} / K_{2}, ρ = r_{2} / r_{1}, γ = 1 - δ / r_{2} .

Each is a ratio of quantities with matching units. In particular

γ

can be negative even though every original model parameter is positive.

The nonnegative quadrant is invariant because both right-hand sides have the corresponding population as a factor. There

u_{2}^{'} \leq ρ γ u_{2}

0 \leq u_{2} (τ) \leq u_{2} (0) e^{ρ γ τ} ⟶ 0 (γ < 0) .

Biologically

γ < 0

δ > r_{2}

: per-capita removal exceeds even the grey squirrels' maximum uncrowded growth rate. Competition can only strengthen the decline.

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