On the Approximation of Physiologically Structured Population Model with a Three Stage-Structured Population Model in a Grazing System

Abstract

A  great deal of ecological theory is based on simple Lotka-Volterra-type of unstructured population models in the study of complex population dynamics and communities. The main reason is to obtain important information for predicting their future evolution. In these unstructured models, it is assumed that all individuals in the population are identical, with the same birth and death rates, and consume equally from shared resources in a homogeneous environment. In reality, these assumptions are not biologically true but still forms a basis for modeling population ecology. We apply this paradigm on the grazing system consisting of coupled ordinary differential equations describing the dynamics of forage resource and livestock population in a grassland ecosystem. We do this by investigating the dynamics of the individuals at different life-history stages of juvenile and adult livestock. The mathematical derivation of the model is carried out to show how the physiologically structured population model can be approximated using a three stage-structured population model. Thus the resulting system of ordinary differential equations can be solved to predict density-dependent properties of the population since it provides a somewhat close-to-reality description of the natural and traditional grazing system. This model therefore certainly contains the needed information in the modeling methodology and accommodates the necessary amount of biological details about the population.

Appendix–Systematic Derivation of the Maturation Rates of Juveniles and Adults

Our task is to write the maturation rates of the juveniles $x_wg_1(x_w,N)l_1(x_w,t)$ and $x_mg_2(x_m,F)l_2(x_m,t)$ in terms of $L_1$, $L_2$ and A.

At the equilibrium, $\frac{\partial }{\partial t}l_1^{*}(x,t) =\frac{\partial }{\partial t}l_2^{*}(x,t)=0$, and the juvenile dynamics becomes

$$\begin{aligned} \frac{\partial }{\partial x}g_1(x,N)l_1^{*}(x,t)= - \mu _1l_1^{*}(x,t). \end{aligned}$$

(34.40)

$$\begin{aligned} \frac{\partial }{\partial x}g_2(x,F)l_2^{*}(x,t)= - (\mu _2 + h_1)l_2^{*}(x,t). \end{aligned}$$

(34.41)

Now by considering Eq. (34.40) and dividing both sides by $g_1(x,N)l_1^{*}(x,t)$ we have;

$$\begin{aligned}\frac{\frac{\partial }{\partial x}g_1(x,N) l_1^{*}(x,t)}{g_1(x,N)l_1^{*}(x,t)} = -\frac{ d_1(F)l_1^{*}(x,t)}{g_1(x,N)l_1^{*}(x,t)} \end{aligned}$$

.

Integrating from size $x_b$ to any size x and using juvenile growth rate we obtain

$$\begin{aligned} \int _{x_b}^{x}\frac{\frac{\partial }{\partial s}g_1(s,N)l_1^{*}(s,t) }{g_1(s,N)l_1^{*}(s,t)}ds= & {} \int _{x_b}^{x}-\frac{ \mu _1l_1^{*}(s,t)}{g_1(s,N)l_1^{*}(s,t)}ds \\ \ln (g_1(s,N)l_1^{*}(s,t))\big |_{x_b}^{x}= & {} - \int _{x_b}^{x} \frac{\mu _1}{g_1(s,F)}ds \\ \ln (g_1(x,N)l_1^{*}(x,t))-\ln (g_1(x_b,N)l_1^{*}(x_b,t))= & {} -\frac{\mu _1}{v_{L_1}}\int _{x_b}^{x}\frac{1}{s}ds \\ \ln \left( \frac{g_1(x,N)l_1^{*}(x,t)}{g_1(x_b,N)l_1^{*}(x_b,t)}\right)= & {} - \frac{\mu _1}{v_{L_1}}\int _{x_b}^{x}\frac{1}{s}ds \\= & {} - \frac{\mu _1}{v_{L_1}}\ln s\bigg |_{x_b}^{x} \\= & {} - \frac{\mu _1}{v_{L_1}}(\ln x-\ln x_b) \\= & {} - \frac{\mu _1}{v_{L_1}}\ln \left( \frac{x}{x_b}\right) \\= & {} \ln \left( \frac{x}{x_b}\right) ^{- \frac{\mu _1}{v_{L_1}}} \end{aligned}$$

or equivalently

$$\begin{aligned} \frac{g_1(x,N)l_1^{*}(x,t)}{g_1(x_b,N)l_1^{*}(x_b,t)}= & {} \left( \frac{x}{x_b}\right) ^{- \frac{\mu _1}{v_{L_1}}} \\ g_1(x,N)l_1^{*}(x,t)= & {} g_1(x_b,N)l_1^{*}(x_b,t) \left( \frac{x}{x_b}\right) ^{- \frac{\mu _1}{v_{L_1}}} \\= & {} g_1(x_b,N)l_1^{*}(x_b,t) x_b^{\frac{\mu _1}{v_{L_1}}}x^{{- \frac{\mu _1}{v_{L_1}}}}. \end{aligned}$$

Therefore the maturation rate of $l_1$ into $l_2$, attained at $x_w$, is given by

$$\begin{aligned} x_wg_1(x_w,N)l_1^{*}(x_w,t) = g_1(x_b,N)l_1^{*} (x_b,t)x_b^{\frac{\mu _1}{v_{L_1}}}x_w^{1-{{\frac{\mu _1}{v_{L_1}}}}}. \end{aligned}$$

(34.42)

The equilibrium juvenile distribution for $l_1$ is therefore given by

$$\begin{aligned} l_1^{*}(x,t)= \frac{g_1(x_b,N)l_1^{*}(x_b,t)}{xv_{L_1}} x_b^{\frac{\mu _1}{v_{L_1}}}x^{{-\frac{\mu _1}{v_{L_1}}}}. \end{aligned}$$

(34.43)

Next we calculate the juvenile biomass density for $L_1$ by substituting Equation (34.2) in Eq. (34.17) so that

$$\begin{aligned} L_1^{*}(t)= & {} \int _{{x_b}}^{x_w}x\frac{g_1(x_b,N)l_1(x_b,t)}{xv_{L_1}} x_b^\frac{\mu _1}{v_{L_1}}x^{-{\frac{\mu _1}{v_{L_1}}}}dx\\= & {} \int _{{x_b}}^{x_w}\frac{g_1(x_b,N)l_1(x_b,t)}{v_{L_1}} x_b^\frac{\mu _1}{v_{L_1}}x^{-{\frac{\mu _1}{v_{L_1}}}}dx\\= & {} \frac{g_1(x_b,N)l_1(x_b,t)}{v_{L_1}}x_b^\frac{\mu _1}{v_{L_1}} \int _{{x_b}}^{x_w}x^{-{\frac{\mu _1}{v_{L_1}}}}dx\\= & {} \frac{g_1(x_b,N)l_1(x_b,t)}{v_{L_1}} x_b^\frac{\mu _1}{v_{L_1}} \frac{x^{1-{\frac{\mu _1}{v_{L_1}}}}}{1-{\frac{\mu _1}{v_{L_1}}}}\bigg |_{x_b} ^{x_w}\\= & {} \frac{g_1(x_b,N)l_1(x_b,t)}{v_{L_1}} x_w^\frac{\mu _1}{v_{L_1}} ~~\frac{x_w^{1-\frac{\mu _1}{v_{L_1}}} - x_b^{1-\frac{\mu _1}{v_{L_1}}}}{1-{\frac{\mu _1}{v_{L_1}}}}\\= & {} \frac{g_1(x_b,N)l_1(x_b,t)}{v_{L_1}}x_b^\frac{\mu _1}{v_{L_1} }v_{L_1}\frac{x_w^{1-\frac{\mu _1}{v_{L_1}}} - x_b^{1-\frac{\mu _1}{v_{L_1}}}}{v_{L_1}-\mu _1}\\= & {} \frac{g_1(x_b,N)l_1(x_b,t)}{v_{L_1}-\mu _1}x_b^\frac{\mu _1}{v_{L_1}} \left( x_w^{1-\frac{\mu _1}{v_{L_1}}}-x_b^{1-\frac{\mu _1}{v_{L_1}}}\right) \end{aligned}$$

or equivalently

$$\begin{aligned}g_1(x_b,N)l_1^{*}(x_b,t)x_b^{\frac{\mu _1}{v_{L_1}}} =\frac{\big (v_{L_1}-\mu _1\big )}{\left( x_{w}^{1-\frac{\mu _1}{v_{L_1}}} - x_{b}^{1-\frac{\mu _1}{v_{L_1}}} \right) }L_1^{*}(t)\end{aligned}$$

and substituting this in Eq. (34.42) we have the juvenile maturation rate of $L_1$ from $x_b$ to $x_w$

$$\begin{aligned} x_wg_1(x_w,N)l_1(x_w,t)= & {} \frac{\left( v_{L_1}- \mu _1\right) }{\left( x_{w}^{1-\frac{\mu }{v_{L_1}}} - x_{b}^{1-\frac{\mu _1}{v_{L_1}}} \right) }L_1^{*}(x,t)x_w^{1-\frac{\mu _1}{v_{L_1}}} \\= & {} \frac{\left( v_{L_1}-\mu _1\right) }{x_w^{1- \frac{\mu _1}{v_{L_1}}} \left( 1-(\frac{x_b}{x_w})^{1-\frac{\mu _1}{v_{L_1}}}\right) } L_1^{*}(x,t)x_w^{1-\frac{\mu _1}{v_{L_1}}}\\= & {} \frac{\left( v_{L_1}-\mu _1\right) }{1- \left( \frac{x_b}{x_w}\right) ^{1-\frac{\mu _1}{v_{L_1}}}}L_1^{*}(t)\\= & {} \gamma (v_{L_1}(F))L_1^{*}(t), \end{aligned}$$

where

$$\begin{aligned} \gamma (v_{L_1})=\frac{\big (v_{L_1}-\mu _1\big )}{1-\left( \frac{x_b}{x_w} \right) ^{1-\frac{\mu _1}{v_{L_1}}}}. \end{aligned}$$

The juvenile maturation rate of $L_2$ from size $x_w$ to size $x_{m}$ can as well be derived in a similar fashion to give

$$\begin{aligned} x_mg_2(x_m,F)l_2(x_m,t) =\gamma (v_{L_2})L_2^{*}(t), \end{aligned}$$

where

$$\begin{aligned} \gamma (v_{L_2})=\frac{v_{L_2}-(\mu _2+h_1)}{1-\left( \frac{x_w}{x_m}\right) ^{1- \frac{\mu _2+h_1}{v_{L_2}}}}. \end{aligned}$$

With these maturation rates for the juveniles substituted in the juvenile dynamics, we obtain the set of ordinary differential equations describing the three stage-structured population model for the grazing system given in (34.36).