Question

A lobster population of size N grows logistically in the absence of fishing. If the fishing fleet is of constant size and has a limited capacity, then the dynamics of the harvested population may be described by the following model: dtdN=RN(1−KN)−B+NAN where R,K,A and B are all positive constants.

(a) If time is measured in years, and the population size is measured in hundreds of tonnes, what are the units of R,K,A and B ?
(b) Show that the model can be expressed in non-dimensional form as dτdn=rn(1−kn)−1+nn stating clearly the definitions of each of n,τ,r and k.
(c) Now assume that k=1/2. Sketch phase lines (non-negative half-line only) and describe the asymptotic behaviour as τ→[infinity] of n(τ) for each of the cases (i) 01.

Answer 1

The units of constants R, K, A, and B in the logistic growth equation are 'per year,' 'hundreds of tonnes,' 'fleet units,' and 'hundreds of tonnes per year,' respectively. The non-dimensional form of the equation helps to analyze population dynamics without specific scales. The asymptotic behavior of the population depends on the relation between r and 1, affecting whether the population declines, stabilizes, or grows indefinitely.

Step-by-step explanation:

The student's question involves determining the units of the constants in the logistic growth equation and expressing the equation in non-dimensional form, followed by analyzing the asymptotic behavior of the equation under a certain condition.

Part (a): Units of Constants

The units for the constants in the logistic growth equation are as follows:

• The growth rate constant R has units of "per year" since it's the rate at which the population grows per unit time.
• The carrying capacity constant K has units of "hundreds of tonnes" because it represents the maximum population size.
• The fishing effort constant A has no clear units given in the problem, but it would typically be some function of the fishing fleet's size or capacity, possibly "fleet units" or similar.
• The catchability coefficient constant B would have units of "hundreds of tonnes per year" as it represents a constant harvest rate.

Part (b): Non-dimensional Form

By defining n = N/K, \tau = Rt, r = R/A, and k = AK/B, we can transform the given logistic growth equation into a non-dimensional form. This transformation simplifies analysis and provides insights into the system's behavior without being tied to specific units or scales.

Part (c): Asymptotic Behaviour

When k = 1/2, the phase lines for the system can be sketched by considering the differential equation. The asymptotic behavior of n(\tau) as \tau approaches infinity will depend on the relationship between r and 1 -- if r is less than, equal to, or greater than 1. This relates to whether the population ultimately decreases to zero, reaches a steady state, or grows without bound.