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By N. Goel (Auth.)

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Example text

We now consider the Verhulst model over all regimes by returning to Eq. 49), which is the Schrödinger equation for the Morse potential if one identifies ft2 with a2/k. The energy levels of a Morse oscillator are known to be En = -A{l-(h2/Ayi2(n+h)}2. , the largest integer less than or equal to x. 49). 44a). 73a) in the regime of the discrete spectrum. Here we have z = qexp(V-V0) with ? 73c) P . ( . ) = E ( · ) ^ , W . / r . 73d) (2 73e) - ™-^ΨΓΨΊΪ> nl T(q— 2n— 1) the definition of (a)» being (a)n=l if n = 0 = α(α+1)···(α+η-1) if n > l .

Let us first consider case (a). 20b) g3 = &i0i/o3i and 04=^2/042. 20c) There are a number of diagrams which express the manner in which graph (a) is applicable. One or more of our four species can feed on the nutrient reservoir with a number of possibilities of whether i eats j , or j eats i. Some of these possibilities are exhibited in Fig. 8. Consider first case (i). We ask for a set of conditions such that each oy>0. Our arrangement of arrows imply 48 NONLINEAR MODELS OF INTERACTING POPULATIONS 2f 4 T 3 T |1 i ί i (i) FIG.

However, to obtain q^s, one must have two species with nonvanishing Verhulst terms. 29) for an even number of species. If only one diagonal term is nonzero, det (an) is still zero because the coefficient of the diagonal term in the expansion of the determinant is an antisymmetric determinant of odd order. 33) for a general system of 2n species with an arbitrary number of species having nonzero Verhulst terms. 57 V. Time-Dependent Fluctuations in Population It was mentioned in the previous section that systems composed of an odd number of species either reach an equilibrium set of populations or decay into systems containing an even number of species.

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