Landscape Diversity Influences Dispersal and Establishment of Pest with Complex Nutritional Ecology

Abstract

We studied the effects of landscape structure on species with resource nutritional partition between the immature and adult stages by investigating how food quality and spatial structure of a landscape may affect the invasion and colonization of the insect pest, Diabrotica speciosa. To this end, we formulated two bidimensional stochastic cellular automata, one for the insect immature stage and the other for the adult stage. The automata are coupled by adult oviposition and emergence. Further, each automata site has a specific culture type, which can affect differently the fitness attributes of immatures and adults, such as mortality, development and oviposition rates. We derived the mean-field approximation for these automata model, from which we obtained conditions for insect invasion. We ran numerical simulations using entomological parameters obtained from laboratory experiments (using bean, soybean, potato, and corn crops), and we compared the results of the automata with the ones given by the mean-field approximation. Finally, using artificially generated landscapes, we discussed how the structured heterogeneous landscape can affect dispersal and establishment of insect populations.

Appendix: Stability of the Equilibria \(P_0\) and \(P_1\)

The stability of an equilibrium $\bar{X}$ of the first-order system of difference equations, $\bar{X}_{n+1} = F(\bar{X}_n),$ is determined by the eigenvalues of the derivative matrix, $J(\bar{X}),$ of $F$ evaluated at $\bar{X}$. If all the eigenvalues, $\lambda _i,$ satisfy $|\lambda _i| < 1$, so that the spectral ratio $r(J) <1$, then $\bar{X}$ is locally asymptotically stable (Allen 2007). For system (1) the derivative $J$ is given by

$$\begin{aligned} J(\rho _1, \rho _2)= \left( \begin{array}{cc} 25 \phi \rho _2 + 1 -\sigma - \mu &{} 25 \phi (1- \rho _1)\\ \displaystyle {\frac{\sigma }{2}}(1-\rho 2) &{} -\displaystyle {\frac{\sigma }{2}}\rho _1 + 1-\gamma \\ \end{array} \right) \end{aligned}$$

(11)

which evaluated in the trivial equilibrium $P_0$, adopts the form

$$\begin{aligned} J(0, 0)=\left( \begin{array}{cc} 1 -\sigma - \mu &{} 25 \phi \\ \displaystyle {\frac{\sigma }{2}} &{} 1-\gamma \\ \end{array} \right) . \end{aligned}$$

(12)

The eigenvalues $\lambda _1, \lambda _2$ satisfy the polynomial

$$\begin{aligned} p(\lambda )=\lambda ^2 + a_1 \lambda + a_2 \end{aligned}$$

with

$$\begin{aligned} a_1&= -\mathrm{Trace\; J} = -(1- \sigma - \mu + 1 - \gamma ), \nonumber \\ a_2&= \mathrm{Determinant\; J} = (1-\mu -\sigma )(1-\gamma ) - \displaystyle {\frac{25 \phi \sigma }{2}}. \end{aligned}$$

(13)

For a two-dimensional system, the condition

$$\begin{aligned} |a_1| < 1 + a_2 < 2 \end{aligned}$$

(14)

assures that $\lambda _1$ and $\lambda _2$ have norm less than one, and therefore the local stability of the equilibrium point (Allen 2007).

Since $\sigma + \mu <1$, and $\gamma <1$, it is easily verified that $a_1$, and $a_2$ satisfy

$$\begin{aligned} |a_1| - (1 + a_2)&= -(\mu + \sigma ) \gamma (1- R_0) \nonumber \\ 1 + a_2&= 2 - (\gamma + \mu + \sigma ) + (\gamma (\mu +\sigma )(1-R_0). \end{aligned}$$

(15)

From these expressions, it follows that inequality (14) holds if and only if $R_0 <1$, and therefore $P_0$ is asymptotically stable for $R_0 <1$.

Next, we analyze the stability of $P_1$. Substituting their coordinates, and replacing $25\phi$ by $\displaystyle {\gamma (\sigma + \mu )\over \sigma } R_0$, $a_1=|-\mathrm{Trace}|$, and $a_2= \mathrm{Determinant}$ of $J(P_1)$ become

$$\begin{aligned} a_1&= (R_0-1)\left[ (\sigma + \mu ) - \displaystyle {\sigma \over 2 R_0}\right] + 1-(\sigma + \mu ) + 1-\gamma , \nonumber \\ a_2&= 1 + \displaystyle {\sigma \over 2}\displaystyle {R_0 -1 \over R_0} \left[ -(\sigma + \mu ) (R_0-1) -(1- 2\sigma - 2\mu )\right] \\&+(\sigma + \mu )(1-\gamma ) (R_0-1) + (1-\gamma ). \nonumber \end{aligned}$$

(16)

For $R_0 =1$, the variables $a_1=2-\sigma -\mu -\gamma$, and $a_2 = 2-\gamma$, which implies that the stability conditions (14) are fulfilled. By continuity, it follows that they are still valid for values of $R_0 >1$ in a neighborhood of $R_0=1$. But since $a_1$ increases without bound when $R_0 \rightarrow \infty$, there must be a value $R_0^*$ such that $a_1 \ge 2,$ and the stability conditions no longer hold for $R_0 > R_0^*$, therefore $P_1$ becomes unstable, and more complicated dynamics arise.