
MBW:A New Model for AgeSize Structure of a PopulationFrom MathBioExecutive SummaryThis paper derives a continuous population model that is dependent on age and mass (or some other behavior defining physiological characteristic). An analytic solution is found, and the model generated is then compared to other existing models. Sinko and Streifer's ModelDefinition of ParametersSinko and Streifer begin by defining as a density function that depends on (time), (age), and (behavior defining physical attribute). The total number of animals between the ages and and the masses and are therefore equivalent to the following: ^{[1]} Relaxing m0 and a0 to zero while also letting m1 and a1 go to infinity will result in the total number of animals in the population. The age and size based model is based on the following partial differential equation: ^{[2]} Where is the growth rate at time for an animal at age with mass , and is the death rate for specimens of age at time with mass . Sinko and Streifer present a lengthy derivation showing that satisfies the above PDE. In order for this problem to be well set (and therefore solvable), boundary conditions must be defined. Specifically,
^{[3]} Where is the number of animals of age and mass that initially exist at time , and is the number of newborn animals with mass that exist at time . is often defined in the following manner: ^{[4]} Where , is the rate at which animals of mass Failed to parse (lexing error): m’ and age give birth to babies of masses to . Analytic SolutionAn analytic solution to the PDE described above exists under special cases. If we assume that is actually a function of and only, and that is independent of , the following analytic solution can be obtained:
^{[5]} This equation completely describes the ages and masses of all animals in the population provided that , , , and are defined. Assuming that and are independent of does restrict this model. It essentially means that the population is not subject to “density effects.”^{[6]} Food limitations and deaths due to increased encounters are density effects that would affect and respectively. As discussed earlier, is also dependent on if the population is generating offspring. If this is the case, the solution found for Sinko and Streifer's PDE is not complete.^{[7]} In the region , the solution is still known, but in the solution , depends on an integral of , which makes the definition of implicit. This makes finding an analytic solution for difficult.^{[8]} Relation to Other ModelsOldfield's ModelD.G.Oldfield (1966) proposed an equation similar to the PDE model presented above. The key differences are that Oldfield’s three variables represent age and two physiological properties. His rate functions are also defined in a way that they only depend on time and the variable that they determine. Following Oldfield’s method will also generate an analytic solution to Sinko and Streifer's model given the following circumstances:^{[9]}
The fourth assumption implies that cannot be mass. Sinko and Streifer note that other analytic solutions probably exist, but that they will most likely place serious restrictions on the population. Sinko and Streifer go on to say that numerical solutions are the only existing means for solving the original PDE given a realistic case. Von Foerster's ModelSinko and Streifer's PDE includes H. Von Forester’s equation (1959) as a special case: ^{[10]} A key difference between Sinko and Streifer's model and Von Foerster's model is that Von Foerster's model is mass independent. Bailey's Parasite/Host ModelVon Foerster's model is the special case of V.A.Bailey's parasite/host model (1931) when there are zero parasites.^{[11]} Sinko and Streifer provide a brief derivation in their paper. Leslie and Lewis's ModelP.H.Leslie (1945) and E.G.Lewis's (1942) population model is a discrete model that depends on time and age. Sinko and Streifer show that as the amount of time in between time steps goes to zero, Leslie and Lewis's model reduces to Von Foerster's model. ^{[12]} Frank's ModelP.W.Frank (1960) used P.H.Leslie’s model with and dependent on the total number of animals in an unsuccessful attempt to describe Daphnia pulex populations.^{[13]} Frank then generalized the model by adding equations to find the mass of the individual, and therefore the biomass of the population. Animal mass is defined by:
^{[14]} Where is the increase in mass in the time for an animal of age in a population with biomass . is defined as: ^{[15]} Sinko and Streifer go on to show that as , Frank's biomass model reduces to their own model. A key difference between the two models is that in Frank's model, any animal of a given age has the same mass, while Sinko and Streifer's model allows for a mass distribution among animals of the same age.^{[16]} Slobodkin ModelL.B.Slobodkin (1935) has also proposed an algebra of population growth in which anmials are divided into different size and age categories. Animals in Slobodkin’s model either grow bigger, stay the same, or die. This is a significant deviation from Sinko and Streifer's model, since in their case animals are allowed to grow or shrink due to environmental factors.^{[17]} Verhulst ModelA very simple logistic model presented by P.F.Verhulst (1838) and L.Pearl and L.J.Reed (1938) can be derived from the Von Foerster model as well. Verhulst's model reads as follows: ^{[18]} Where is the intrinsic rate of increase, and is the equilibrium population size. is the population size at time . One can quickly infer from this model that population growth/decay is a function of density only. In the process of showing that Verhulst's model is a special case of Von Foerster's model, Sinko and Streifer also show that Verhulst's model assumes animals of all ages die at the same rate, and that birth rate is independent of animal density.^{[19]} References
