Write the difference equation describing the population at time "t" in terms of time. This equation is N(t + 1) = exp(-M)*N(t) – g(M)C(t). In this equation, N(t) describes the number of fish at time t, “exp” represents the exponential function, M is the fish mortality, g(M) is the ratio of adjusted catch to true catch and C(t) is the catch at time t.
Write the standard equation that describes the change in population of fish. This equation is N(t + 1) = N(t)*exp(-F – M). In this equation “F” represents the mortality due to fisheries.
Combine the two equations. Notice that both equations have the same left side. Thus, you can equate the right sides of both equations, yielding exp(-M)*N(t) – g(M)C(t) = N(t)*exp(-F – M).
Simplify the new equation. Move the terms with N(t) to the same side of the equation side, using algebra. This results in g(M)C(t) = N(t)*{exp(-M) – exp(-F – M)}.
Solve for C(t). This yields C(t) = N(t)*{exp(-M) – exp(- F – M)}/g(M).
Replace g(M) with its mathematical definition, (F + M)*[exp(-M) – exp(-F – M)]/[F*(1 – exp(-F – M))].
Simplify. After like-terms cancel, the result, Baranov’s catch equation, is C(t) = F*(1 – exp(-F – M)*N(t)/[F + M].