Agricultural landscape structure and invasive species: the cost-effective level of crop field clustering

Abstract

Invasive pests in agricultural settings may have severe consequences for agricultural production, reducing yields and the value of crops. Once an invader population has established, controlling it tends to be very expensive. Therefore, when the potential impacts on production may be great, protection against initial establishment is often perceived to be the most cost-effective measure. Increasing attention in the ecological literature is being given to the possibility of curbing invasion processes by manipulating the field and cropping patterns in agricultural landscapes, so that they are less conducive to the spread of pests. However, the economic implications of such interventions have received far less attention. This paper uses a stochastic spatial model to identify the key processes that influence the vulnerability of a fragmented agricultural landscape to pests. We explore the interaction between the divergent forces of ecological invasion pressure and economic returns to scale, in relation to the level of clustering of crop fields. Results show that the most cost-effective distances between crop fields in terms of reducing food production impacts from an invasive pest are determined by a delicate balance of these two forces and depend on the values of the ecological and economic parameters involved. If agricultural productivity declines slowly with increasing distance between fields and the dispersal range of the potential invader is high, manipulation of cropping structure has the potential to protect against invasion outbreaks and the farmer can gain benefit overall from maintaining greater distances between fields of similar crops.

Appendix: Calculation of the probability of a field being occupied by the pest species

The metapopulation dynamics is modelled as a Markov process for a 2-field system. We denote as p ₀₀, p ₁₀, p ₀₁ and p ₁₁ the probabilities of both fields being empty (not occupied by the pest species), respectively field 1 being occupied and field 2 being empty, respectively field 1 being empty and field 2 being occupied, respectively both fields being occupied. Assuming that only one transition can occur at a given point in time, these probabilities change in time via

$$ \begin{array}{l}{\overset{\cdot }{p}}_{00}=-2i{p}_{00}+e\left({p}_{10}+{p}_{01}\right)\hfill \\ {}{\overset{\cdot }{p}}_{10}=i{p}_{00}-\left(i+e+c\kern.18em \exp \left(-\alpha d\right)\right){p}_{10}+e{p}_{11}\hfill \\ {}{\overset{\cdot }{p}}_{01}=i{p}_{00}-\left(i+e+c\kern.18em \exp \left(-\alpha d\right)\right){p}_{01}+e{p}_{11}\hfill \\ {}{\overset{\cdot }{p}}_{11}=\left(i+e+c\kern.18em \exp \left(-\alpha d\right)\right)\left({p}_{10}+{p}_{01}\right)-2e{p}_{11}\hfill \end{array} $$

(8)

Considering, e.g., the first equation, the first term on the r.h.s. considers colonisation of field 1 or field 2 from outside the system and the second term the extinction of a population on field 1 respectively 2. The middle term, on the r.h.s. of the second equation, considers colonisation of field 2 from outside, extinction of the population on field 1, and colonisation of field 2 by emigrants from field 1.

We are interested in the steady state, ${\overset{\cdot }{p}}_{00}={\overset{\cdot }{p}}_{10}={\overset{\cdot }{p}}_{01}={\overset{\cdot }{p}}_{11}=0$ of the system, which is obtained with some algebra as

$$ \begin{array}{l}{\widehat{p}}_{00}=\frac{e}{i}{\left(2+\frac{e}{i}+\frac{i+c \exp \left(-\upalpha \mathrm{d}\right)}{e}\right)}^{-1}\kern1em \\ {}{\widehat{p}}_{10}={\widehat{p}}_{01}={\left(2+\frac{e}{i}+\frac{i+c \exp \left(-\upalpha \mathrm{d}\right)}{e}\right)}^{-1}\kern1em \\ {}{\widehat{p}}_{11}=1-{\widehat{p}}_{00}-{\widehat{p}}_{10}-{\widehat{p}}_{01}\kern1em \end{array} $$

(9)

The probability of field 1 respectively field 2 being occupied by the pest species is ${p}_1={\widehat{p}}_{10}+{\widehat{p}}_{11}$ respectively ${p}_2={\widehat{p}}_{01}+{\widehat{p}}_{11}$. Inserting Eq. (9) delivers Eq. (3).

For N = 3 or more fields the steady state cannot be determined analytically. Instead the metapopulation dynamics on the fields is simulated. For this we form all possible states from p _00…0 to p _11…1 where each index refers to one of the fields and is 0 if the field is empty and 1 if it is occupied by the pest species. The transitions between all 2^N states are determined analogously to Eq. (8), leading a set of to 2^N differential equation dp…/dt = … Each equation is discretised with respect to time by replacing dp/dt by [p(t + Δt)-p(t)]/Δt with sufficiently small time interval Δt. For Eq. (8), e.g., one would obtain

$$ \begin{array}{l}{p}_{00}\left(t+\varDelta t\right)={p}_{00}(t)-\left\{2i{p}_{00}+e\left({p}_{10}+{p}_{01}\right)\right\}\varDelta t\hfill \\ {}{p}_{10}\left(t+\varDelta t\right)={p}_{10}(t)+\left\{i{p}_{00}-\left(i+e+c \exp \left(-\alpha d\right)\right){p}_{10}+e{p}_{11}\right\}\varDelta t\hfill \\ {}{p}_{01}\left(t+\varDelta t\right)={p}_{01}(t)+\left\{i{p}_{00}-\left(i+e+c \exp \left(-\alpha d\right)\right){p}_{01}+e{p}_{11}\right\}\varDelta t\hfill \\ {}{p}_{11}\left(t+\varDelta t\right)={p}_{11}(t)+\left\{\left(i+e+c \exp \left(-\alpha d\right)\right)\left({p}_{10}+{p}_{01}\right)-2e{p}_{11}\right\}\varDelta t\hfill \end{array} $$

(10)

Starting from the (without loss of generality) initial condition p _00…0(t = 0) = 1 and zero probability for all other states, we determine the probabilities p(Δt) for all states, insert these probabilities into the equation system to determine the probabilities for time t = 2Δt, and so on, until the unique steady state is reached where the probabilities do not change any more. The probability of observing field i occupied then is the sum the probabilities of all states in which field i is occupied.

Inserting these probabilities into Eq. (4) leads to Eq. (6) for two fields. The first order necessary condition is given by

$$ \begin{array}{l}\frac{dB}{dd}=\frac{-2}{\left(1+i/e\right){\left(1+k \exp \left(-\alpha d\right)\right)}^2}\times \hfill \\ {}\kern2.5em \left[\beta w \exp \left(-\beta d\right)\left(1+k \exp \left(-\alpha d\right)\right)+\alpha k \exp \left(-\alpha d\right)\left(1+w \exp \left(-\beta d\right)\right)\right]=0\hfill \end{array} $$