Solve and interpret continuous logistic models

Unit 4 modelling task 1: define a candidate model

A controlled culture has P in biomass units and t in days. The coefficients are specified independently for this exercise, not fitted from the later validation sample. Interpret the model as a deterministic candidate under fixed conditions, with carrying capacity 80 and initial relative growth rate reduced by crowding

What is the initial growth rate in biomass units per day?

Derive and investigate the model, ungraded

Solve the initial-value problem, state the physical time domain and range, identify the maximum growth rate and its time, and compare the model with a constant-relative-growth alternative. Explain which observations and assumptions would be needed to defend either model

Model: separate, retain the initial condition and check

For 0<P<80, separation gives ln[P/(80−P)]=0.4t+C. P(0)=20 gives eᶜ=1/3, so P(t)=80/(1+3e^(−0.4t)). Substitution and the initial condition verify it. For physical t≥0, 20≤P<80. The rate 0.4P(1−P/80) is maximal at P=40, with rate 8, reached at T=2.5ln3 days. The mathematical formula is defined for all real t, but the stated observation model uses t≥0. The constant-relative-growth model E′=0.4E would start at rate 8, so it is not a like-for-like initial-rate match

At what biomass, in biomass units, is growth fastest?

What is the maximum growth rate in biomass units per day?

Vary the growth assumption

To match the initial value and initial rate, use E(t)=20e^(0.3t), since E′(0)=6. Its unbounded growth differs from the logistic saturation. At the logistic half-capacity time T=2.5ln3, E(T)=20×3^(3/4), about 45.59, whereas P(T)=40. Agreement at the starting value and slope does not determine later behaviour

Which evidence could distinguish the logistic and constant-relative-growth models?

Model critique and reporting guide

A full report should identify fixed environment, a homogeneous aggregate population, capacity and rate assumptions, measurement error and the range of valid extrapolation. Proposed changes such as time-varying capacity or removal alter the differential equation. Without suitable data the algebra verifies a solution of the assumed equation, not the culture model itself