Ecological Modelling 306 (2015) 240 246 Contents lists available at ScienceDirect Ecological Modelling journal homepage: www.elsevier.com/locate/ecolmodel The role of heterogeneous agricultural landscapes in the suppression of pest species following random walk dispersal patterns L. Potgieter a,, J.H. van Vuuren b, D.E. Conlong c,d a Department of Logistics, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa b Department of Industrial Engineering, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa c Department of Conservation Ecology and Entomology, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa d South African Sugarcane Research Institute, Private Bag X02, Mount Edgecombe, KwaZulu-Natal 4300, South Africa article info abstract Article history: Available online 12 December 2014 Keywords: Heterogenous landscape Reaction diffusion model Agricultural crop Pest suppression Harvesting The contribution of a heterogeneous agricultural landscape (in terms of field layouts of diversified crop ages) towards pest suppression is considered in this study. The population dynamics of a pest species is modelled for a number of different mosaic configurations of differently aged agricultural crops across a spatial domain, where the harvesting of these fields occurs at different points in time. A reaction diffusion model (with constant diffusion coefficients) of the pest species growth and dispersal within a temporally variable and spatially heterogeneous environment is utilised to perform simulations. The primary objective is to establish whether or not there exist certain field configurations of differently aged crops in which average infestation levels are a minimum. It is found that more diversified field configurations (in terms of crop age) with the total length of boundaries shared between differently aged crops at a minimum, yield lower average infestation levels. 2014 Elsevier B.V. All rights reserved. 1. Introduction It has long been recognised that the manipulation of planting and harvesting schedules may impact negatively on pest populations in agricultural crops (Anon., 1996). Mobile organisms are expected to adjust foraging decisions to the local distribution and quality of resources. In the case of pest insects, resource availability is often linked to changes during the life cycle of their host plants. The plant growth stage is directly related to its suitability as a host (quality of resources) and may, in turn, have an influence on infestation levels and movement patterns (Mazzi and Dorn, 2012). Infestation levels decrease to zero or almost zero in crop plants when harvested the pests in immobile or semi-mobile life stages, such as pupae and larvae, are removed together with the harvested crop and those in mobile life stages adjust their foraging, mating and oviposition decisions to the local distribution and quality of remaining host plant resources. Due to harvesting being considered as such a large controlling factor of pest species in crop plants, the question has been raised whether there exist suitably diversified agricultural landscapes (with respect to crop age), in which the Corresponding author. Tel.: +27 21 808 2492. E-mail addresses: lpotgieter@sun.ac.za (L. Potgieter), vuuren@sun.ac.za (J.H. van Vuuren), Des.Conlong@sugar.org.za (D.E. Conlong). harvesting of the different fields at different points in time impact negatively on pest populations. If this question can be answered in the affirmative, then a combination of such field layouts with other pest control methods may significantly reduce the cost of achieving suppression. The primary objective of this paper is, therefore, to examine, using mathematical simulation models, whether or not there exist certain heterogeneous spatial configurations of agricultural landscapes (with respect to fields with differently aged crops), in which harvesting of these fields at different time points yield lower average infestation levels than in other spatial configurations. The population dynamics of the pest species Eldana saccharina Walker (Lepidoptera: Pyralidae) in sugarcane (Saccharum officinarum L.) is taken as application scenario in this study, with the model presented in this paper similar to the models discussed and validated by Potgieter et al. (2012, 2013), but excluding the effect of the sterile insect technique. 2. Description of host plant and pest species Sugarcane is a tall perennial grass from the family Gramineae. To ensure the next year s crop, the sugarcane roots are left in the field to allow shoots to regrow for an additional year (a ratoon) when the stalks are harvested. When the fields are replanted, the roots are ploughed out and mature sugarcane stalks are cut into http://dx.doi.org/10.1016/j.ecolmodel.2014.11.029 0304-3800/ 2014 Elsevier B.V. All rights reserved.
L. Potgieter et al. / Ecological Modelling 306 (2015) 240 246 241 sections (setts), laid horizontally in furrows and covered with soil. Because of the long harvesting season, sugarcane farmers often have fields of different ages on their farms. These are also made up of different varieties bred for specific agronomic conditions. The maturation time of sugarcane varies according to the location. In South Africa, the maturation time for sugarcane varies between 12 and 24 months. During this period various diseases and pests can damage sugarcane which, in turn, may result in yield losses. One of the most serious pest species in sugarcane is the stalk borer E. saccharina. Various methods have been proposed for the control of E. saccharina in sugarcane (Carnegie, 1981; Heathcote, 1984; Conlong, 2001; Keeping and Rutherford, 2004; Webster et al., 2005). Although contributing towards suppression, none of these have proven successful in preventing E. saccharina infestations. Harvesting of sugarcane at an early age is still considered an important controlling factor for E. saccharina in sugarcane (Atkinson and Carnegie, 1989). The lifecycle of E. saccharina is typical of insects consisting of eggs, larvae, pupae and moths. The time spent in each stage of the lifecycle is variable, depending as much on the quality of food as on the temperature experienced. As a result of individual variation in development time and the longevity of the crop, generations completely overlap with all stages present in the crop at all times (Atkinson and Carnegie, 1989). Adult females lay their eggs in hidden positions on the dead leaves and sheaths on the lower parts of the plant. Larvae hatching from eggs feed initially on organic matter under leaf sheaths. When the larvae are sufficiently robust, they start boring into the stalk, and feed on the internal tissues causing yield losses in sugarcane (Atkinson, 1979; Carnegie, 1974). E. saccharina seems to be a relatively weak flier (Atkinson and Carnegie, 1989). However, research results suggest that a small proportion of individuals may disperse over longer distances. The female, being larger and stronger than the male, is thought to be the more likely to migrate (Atkinson, 1981). 3. The model An E. saccharina population is considered in its various stages within a closed spatial domain, which, in the context of this study, is assumed to be an isolated set of adjacent sugarcane fields with heterogeneous ages, surrounded by land uses not considered as possible habitats for E. saccharina. A population growth and dispersal model in a spatially and temporally variable environment is used to describe the spatial dynamics of the set of interacting subpopulations in this isolated and bounded spatial domain. Consider the various stages of an E. saccharina population within a closed, simply connected, two-dimensional spatial domain S. Let E 1 (,t), E 2 (,t), E 3 (,t), E 4 (,t) and E 5 (,t) denote the densities (measured in e/100s, which denotes the number of members of the subpopulation per 100 stalks) of the five subpopulations of eggs, larvae outside the sugarcane stalks, larvae inside the sugarcane stalks, pupae, and moths, respectively, at position = [ 1, 2 ] T S and at time t [0, ). Not much is known about the dispersal patterns of E. saccharina moths except that they are weak fliers. In this study, adult moths are assumed to have no spatial working memory; individual movement is rather represented by a pure random walk. If individual moths follow a pure random walk, population movement may be approximated by a pure diffusion process (Ovaskainen, 2008). Also, a diffusion process is a good approximation in the case of weak fliers. Larval dispersal is not considered applicable since the distances they cover are very small compared to the spatial scale of the model. The diffusion matrix, D, therefore has zero entries corresponding to all subpopulations, except for moths. That is, D = diag {0, 0, 0, 0, d}, where the (constant) diffusion coefficient d is a measure of how effectively moths disperse between habitat sites. The dynamics of the E. saccharina subpopulations in the various stages are described by the reaction diffusion equation E(,t) t = f (,t,e) + D 2 E(,t) (1) under these assumptions, where E(,t) = [E 1 (,t),...,e 5 (,t)] T and where f (,t,e) = [f 1 (,t,e),...,f 5 (,t,e)] T and :=( / 1, / 2 ). The i-th entry of f (,t,e) denotes the number of individuals of the i-th subpopulation created during time t. More specifically, f 1 (,t,e) = 0.5E 5 (,t) ( E (t, ) + E(t, ))E 1 (,t), f 2 (,t,e) = E(t, )E 1 (,t) ( L1 (,t,) + L1 (t, ))E 2 (,t), f 3 (,t,e) = L1 (t, )E 2 (,t) ( L2 (,t,) ( 1 + b(,t)((e 3 )(,t)) ) + L2 (t, ))E 3 (,t), f 4 (,t,e) = L2 (t, )E 3 (,t) ( P (t, ) + P(t, ))E 4 (,t), f 5 (,t,e) = P(t, )E 4 (,t) M (t, )E 5 (,t), where denotes the egg laying rate of a female, E (t, ), P (t, ) and M (t, ) denote the egg, pupal and moth stage-specific mortality rates at time t at a temperature of degrees, L1 (,t,) and L2 (,t,) denote the larval mortality rates outside and inside the stalks at position, at time t and at a temperature of degrees. Furthermore, b(,t) denotes the density-dependent mortality parameter at position and time t, and E(t, ), L1 (t, ), L2 (t, ) and P(t, ) denote the egg, external larval, internal larval and pupal maturation rates at time t and at a temperature of degrees, respectively. A finite difference approximation is used to solve Eq. (1) numerically. 3.1. Boundary conditions Zero-flux Neumann boundary conditions of the form E( 1, 2,t) 1 = 0 and E( 1, 2,t) S 2 = 0 (2) S are assumed for Eq. (1), where S is the boundary of S. Assuming the boundary is not part of S, boundary values for E and D on the discretized domain are similar to those described by Potgieter et al. (2013). 3.2. Initial values Three possible initial infestations are assumed in this study, namely 0% across the entire newly harvested/planted area, low levels (0.1 e/100s) at a small percentage (1%) of randomly selected points (with 0 e/100s at the other points), and low levels (0.1 e/100s) uniformly spread across the entire (100%) harvested area. A newly planted field and a newly harvested field are assumed to have the same initial infestation. Assuming no initial infestation (0%) after harvesting (i.e. the E. saccharina individuals in all life stages are removed during harvesting) is similar to a scenario where infestation only occurs in the form of diffusion from neighbouring infested areas (E. saccharina gradually spreading from the edges inwards). Assuming that infestation occurs only at randomly selected points in a newly planted/harvested area, corresponds to a scenario where not all E. saccharina are removed at the time of harvest, where some of the planted seed cane is infested with E. saccharina, or where some moths not only spread gradually along the edges of fields, but also fly longer distances to infest neighbouring fields within-field. The initial infestation levels of older sugarcane fields at the start of simulations are estimated by the mean-field model of Potgieter et al. (2012).
242 L. Potgieter et al. / Ecological Modelling 306 (2015) 240 246 Fig. 1. Spatial patterns formed by differently aged fields (each crop age is represented by a unique colour) in the spatial domain. (a) A random segmented pattern, (b) two age categories distributed in lanes, (c) three age categories distributed in lanes, (d) two age categories distributed in a checkerboard pattern, (e) three age categories distributed along diagonals in a checkerboard pattern, (f) three age categories distributed in an alternative checkerboard pattern. These patterns, as well as the dimensions, of the fields are varied in the simulations. 3.3. Growth, maturation and mortality parameters Growth, maturation and mortality parameters are assumed similar to those used in the study by Potgieter et al. (2012). Each life stage of E. saccharina has unique maturation and mortality rates which depend on the daily average temperature (Way, 1995). The mortality and maturation rates of each life stage at time t are determined by finding the lowest degree polynomials which achieve a satisfactory fit to the corresponding stage mortality data. Increasing crop age results in higher nutrition levels for E. saccharina larvae, which increases the carrying capacity of the crop. This is modelled by assuming the decreasing s-shaped densitydependent mortality function y b(t) = [a(t)] z + 1, (3) where y >0,z > 1 and a(t) denotes the age of the crop in days at time t (Potgieter et al., 2012). 3.4. Harvesting assumptions The harvesting of sugarcane is assumed to occur at the end of the cycle of 12 or 24 months. Harvesting and planting times are selected only between March and November so as to prevent harvesting occurring during the period between November and March when the mills are closed. 3.5. Heterogeneous landscape structures A rectangular spatial domain S is assumed, which is partitioned into smaller rectangular or square fields S i that are heterogeneous in terms of crop age. They are assumed to be arranged according to the spatial patterns shown in Fig. 1. The average E. saccharina infestation levels over a set period of time are compared for these different patterns in order to examine whether the average infestation level over time is reduced as a result of certain field layout structures. (The Mathworks Inc, 2012) on the differently structured heterogeneous domains, where harvesting occurs at different points in time for the differently aged crops in the domain. Homogenous domains were excluded from the simulations due to the high risk associated with no diversification if, for example, a natural disaster occurs, a farmer may loose his entire crop with no income in that particular sugarcane cycle. A constant diffusion (dispersal) rate of 0.025 is assumed, defined as the number of moths that diffuse (disperse) across a unit area of 25 m 2 per day from a certain position in the spatial domain. The value of the diffusion coefficient in this study was chosen according to the range of 0.005 < d(x) < 0.03 recommended by Potgieter et al. (2013). Simulations were performed over a 4-ha (200 m 200 m) domain over a time period of 84 months. Monthly averages for temperatures in KwaZulu-Natal, South Africa, where sugarcane is grown, were used in the simulations, i.e. temperature was not taken constant during the simulations, but variable according to the month of the year. The density-dependent mortality function b(t) = 4/([a(t)] 2.5 + 1) was assumed. During each time step, the mean of the infestation level 1 across the heterogeneous domain was computed from an infestation surface such as that in Fig. 2. An example of the mean infestation across the entire domain, computed over all time steps, is given in Fig. 3. Different heterogeneous domains were compared using the mean of the infestation level over time (the dotted line in Fig. 3), denoted by E mean, and the standard deviation from this mean, denoted by E stdev, as simulation performance measures. Initial infestation levels include 0%, 1% and 100% for all simulations. Initial infestation levels of 10%, 40% and 70% were, however, also included to compare results with the two extremes (0% and 100%). From Tables 1 and 2, it is clear that E mean is lower for a heterogeneous domain in which fields are larger (e.g. 100 m 100 m), compared to smaller fields (e.g. 40 m 40 m). This difference is more pronounced when assuming that the initial infestation is mainly as a result of diffusion from neighbouring fields (spreading from the edges inwards). At a 100% initial infestation (i.e. a homogenous initial infestation greater than zero across the entire field), the differences in E mean are small. At initial infestation levels 4. Simulation results Average infestation levels over time were estimated by a finitedifference approximation of (1) over time, implemented in Matlab 1 Only the number of larvae and pupae inside the stalk is reported on in this paper, as this is the industry measure used to determine the severeness of an E. saccharina infestation in sugarcane. The infestation level is given as the combined number of larvae and pupae per 100 sugarcane stalks, i.e. e/100s.
L. Potgieter et al. / Ecological Modelling 306 (2015) 240 246 243 Fig. 2. Infestation level (e/100s) at a single time step across a 200 m 200 m heterogeneous spatial domain partitioned into fields containing three different crop ages according to the checkerboard pattern in Fig. 1. Fig. 3. The mean infestation level (e/100s) over a heterogeneous spatial domain partitioned into fields containing three different crop ages, computed over a time period of 7 years. Table 1 E mean (e/100s) for the spatial patterns depicted in Fig. 1. The crop ages were chosen such that, if only two ages were selected, the ages were 12 months apart for two-year cycle sugarcane, while if three ages were selected, the ages were 8 months apart. Pattern Field dimension (in m) Initial infestation 2 ages 3 ages 2 ages 3 ages 2 ages 3 ages Checker 1 40 40 22.47 20.61 22.62 20.75 24.04 22.33 (Fig. 1(d,e)) 100 100 13.63 12.31 17.55 15.47 23.91 22.1 Checker 2 40 40 N/A 21.01 N/A 21.17 N/A 22.73 (Fig. 1(f)) 40 100 N/A 19.68 N/A 20.2 N/A 22.63 Lanes 40 200 20.05 19.07 20.73 19.53 23.7 22.45 (Fig. 1(b,c)) 50 200 18.78 16.84 19.71 18.03 23.98 22.22 100 200 9.24 N/A 15.94 N/A 23.87 N/A Random 40 40 17.94 19.00 19.69 22.32 (Fig. 1(a)) 100 100 9.34 between 0% and 100%, the value of E mean lies between that of a 0% and 100% initial infestation. From Table 1, it can also be seen that the value E mean is slightly lower for a heterogeneous domain with three crop ages than for a domain with only two crop ages present. This difference is not considerably influenced by the choice of initial infestation level. Furthermore, E mean is lower for a heterogeneous domain with fields of different crop ages arranged in lanes (patterns b or c in Fig. 1) compared to fields arranged in a Table 2 E mean (e/100s) for the spatial patterns depicted in Fig. 1(a and b) with initial infestation levels of 10%, 40% and 70%. The crop ages were chosen such that, if only two ages were selected, the ages were 12 months apart for two-year cycle sugarcane, while if three ages were selected, the ages were 8 months apart. The E mean values for the spatial patterns depicted in Fig. 1(a, d and e) and f) with initial infestation levels of 10%, 40% and 70% also lie between the E mean values of the two extremes (0% and 100% initial infestation). Pattern Field dimension (in m) Initial infestation 10% 40% 70% 2 ages 3 ages 2 ages 3 ages 2 ages 3 ages Lanes 40 200 22.11 20.88 23.07 21.81 23.45 22.22 (Fig. 1(b,c)) 50 200 22.19 20.30 23.3 21.48 23.72 21.93 100 200 21.45 N/A 23.1 N/A 23.58 N/A
244 L. Potgieter et al. / Ecological Modelling 306 (2015) 240 246 Fig. 4. A 4-ha (200 m 200 m) domain partitioned into four fields, with three crop ages present. (a) E mean = 11.71 e/100s if fields are chosen as squares of dimensions 100 m 100 m and assuming a 0% initial infestation after harvesting. b) E mean = 15.05 e/100s if fields are chosen as rectangles of dimensions 50 m 200 m and assuming a 0% initial infestation after harvesting. checkerboard pattern (patterns d, e or f in Fig. 1). Finally, domains in which fields form random spatial patterns yield a lower value of E mean than domains in which fields are grouped according to a more organised pattern. Taking all of the above observations into account, it is clear that when assuming that individual moths follow a pure random walk, a domain in which same-aged crops are grouped together (i.e. n fields for n crop ages, with n > 2), results in a lower value of E mean than domains in which the opposite is true (i.e. same-aged crops are separated and scattered across the domain according to some pattern). This observation is also correlated by the conclusion that random patterns perform better, since more same-aged crops are thus expected to be grouped together. Partitioning the domain into n fields for n crop ages may be performed in a number of different ways. The partition of the spatial domain in terms of surface area occupied by the different crop ages, the choice of crop ages, and the order in which the fields of different crop ages are arranged next to one another, were also considered, as described in the following subsections. 4.1. Boundaries shared between subsets An optimal partition of the spatial domain into n fields containing n different crop ages is one for which the expression n f (S i ) (4) i is minimised, where the function f (S i ) denotes the total length of boundaries shared between the field S i and other fields. Minimising the total length of boundaries shared between fields containing different crop ages minimises the cross-infestation. An example of this phenomenon is illustrated in Fig. 4, where a 4 ha domain is partitioned into four fields of equal surface area in two different ways. The partition minimising the length of the boundaries shared between the fields achieved the lowest infestation levels at the end of the simulation. Fig. 5. A 4-ha (200 m 200 m) domain divided into four fields, with four different crop ages present. (a) Fields are chosen as squares of dimensions 100 m 100 m. (b) Fields chosen with unequal surface areas. 4.2. Surface area allocation to fields If a square spatial domain is partitioned into four fields containing four crop ages, the expression in (4) is minimised by partitioning the domain as illustrated in both Fig. 5(a and b), with the difference in surface areas of the fields additionally minimised in (a). The value of E mean does not differ significantly for the domains in (a) and (b). The standard deviation, E stdev, on the other hand, is minimised if the difference between the surface areas of different fields is minimised, which is consistent with the simulation results observed for three fields containing three different crop ages. Although the difference in surface areas does not considerably influence the value of E mean, it does considerably influence the value of E stdev (see Table 3). A smaller standard deviation is considered better, since the height of the peaks (periods of high infestation) in the average infestation across the domain will be smaller. Minimising the difference in surface areas of the various fields is therefore considered better. 4.3. Variation in crop age For 0% initial infestation after harvesting, a 200 m 200 m spatial domain containing only two crop ages, partitioned into two fields of equal surface area, yields a lower value for E mean than a similar domain partitioned into three or more fields containing three or more crop ages (see Table 4). This difference becomes small if a small percentage of initial infestation is allowed, and for 100% initial infestation, a domain containing more crop ages yields only a slightly lower value for E mean. Since an initial infestation greater than 0% is more likely to occur, a more diversified domain in terms of crop age is considered to be better. The value of E stdev is also reduced if more crop ages are present over the spatial domain. A 200 m 200 m domain containing four crop ages performs slightly worse in terms of mean infestation levels than a domain containing three crop ages, but the corresponding value of E stdev is slightly lower. A 200 m 200 m domain containing six crop ages performs considerably worse in terms of mean infestation levels than a domain containing two, three or four crop ages, but the corresponding value of E stdev is much lower as the percentage of initial infestation becomes larger. This seems to contradict the observation that more diversification in terms of crop age is better. Table 3 The values of E mean and E stdev for a 4-ha domain partitioned into three or four fields with equal or unequal surface area allocation to different crop ages. Number of crop ages Surface area Initial infestation 3 Equal 10.46 6.43 14.28 8.64 19.71 10.92 3 Unequal 10.15 7.24 13.77 8.84 19.98 11.31 4 Equal 11.64 6.61 14.67 8.14 20.06 10.65 4 Unequal 11.46 7.03 14.44 9.47 19.80 11.72
L. Potgieter et al. / Ecological Modelling 306 (2015) 240 246 245 Table 4 The values of E mean and E stdev for a 4-ha domain partitioned into two, three, four or six fields of equal area, each with a different crop age. Number of crop ages Initial infestation 2 8.04 7.83 14.25 12.95 21.34 16.3 3 10.46 6.43 14.28 8.64 19.71 10.92 4 11.64 6.61 14.67 8.14 20.06 10.65 6 14.42 6.86 16.07 7.77 19.95 9.62 Fig. 6. A 4-ha (200 m 200 m) domain partitioned into four fields, with four different crop ages present. (a) Variation 1. (b) Variation 2. (c) Variation 3. The four gray scales from the top left to bottom right represent the sugarcanes of 0, 6, 12, and 18 months old, respectively. Table 5 The values of E mean and E stdev for a 4-ha domain partitioned into four fields, with different age arrangements, as illustrated in Fig. 6. Variation Initial infestation 1 11.64 6.61 14.67 8.14 20.06 10.65 2 11.63 6.65 15.34 8.59 20.07 10.66 3 11.63 6.63 14.72 8.27 20.06 10.66 Table 6 The values of E mean and E stdev for a 4-ha domain partitioned into three fields with a different choice of crop ages. Crop age (in months) Initial infestation 0, 8, 16 10.46 6.43 14.28 8.64 19.71 10.92 0, 4, 18 11.42 9.04 14.85 11.41 20.40 13.74 Closer examination of this phenomenon revealed, however, that the size of the domain influences the optimal number of crop ages. This is consistent with the results reported in Table 1, which indicate that larger fields perform better than smaller fields. A 200 m 200 m domain is optimally partitioned into three fields containing three different crop ages. For a domain larger than 4 ha, more than three crop ages may be better since the size of the fields may be larger. Each differently sized and shaped domain, therefore, induces a different optimal partition, depending on both the length of boundaries shared between the fields and the surface areas allocated to the different fields. Fields containing different crop ages may be arranged in many different ways within a certain spatial domain. The three different ways in which four equally sized fields containing different crop ages may be arranged in a square domain are illustrated in Fig. 6. Simulations indicate that there is no significant difference in the resulting values of E mean and E stdev between the three different arrangements (see Table 5). However, minimising the age difference between neighbouring fields (i.e. placing younger crops in close proximity to other younger crops), as in Fig. 6(a), yields a slightly lower value for both E mean and E stdev. The choice of crop ages in a domain may also influence the values of E mean and E stdev. As may be seen from Table 6, at 0% initial infestation after harvesting, a 200 m 200 m domain containing three crop ages, namely, 0, 8 and 16 months old, grouped into three fields, yields a slightly lower value for E mean than a similar domain partitioned into three fields containing crops of ages 0, 4 and 18 months. The value of E stdev, however, is much lower in the former case. This is consistent with the results reported in Table 4, since if crop ages are not uniformly spread over the sugarcane cycle, with two very similar crop ages occurring, the results will closely resemble those of a simulation with only two crop ages. It is expected that the values of E mean and E stdev should lie somewhere between those for a domain with only two crop ages, and a domain with three crop ages uniformly spread over the sugarcane cycle. 5. Conclusion The explicit consideration of pest dispersal within habitat management programmes is relatively recent. Mazzi and Dorn (2012)
246 L. Potgieter et al. / Ecological Modelling 306 (2015) 240 246 highlighted the fact that the success of environmentally sound control practices hinges on a pest species dispersal performance. Experimental evidence points to heterogeneous sensitivity in herbivorous insects, but each species reacts differently according to their dispersal performance within a specific landscape structure. Mean infestation levels in different heterogeneous agricultural landscapes (with respect to crop age) were considered in this study for pest species exhibiting pure random walk dispersal patterns. Simulation results suggest that an optimal partition of the landscape in terms of crop age exists, minimising mean infestation levels across the domain. This optimal structure may be combined with other pest control programmes to improve the cost-efficiency thereof. The results obtained in this study indicate that same-aged patches should all be grouped together into larger fields in some optimal arrangement, where boundaries shared between differently aged fields as well as the difference in surface areas of the fields are minimised, in order to minimise mean infestation levels across the domain. The results presented here are, however, subject to the assumption of constant diffusion coefficients, which may be an unrealistic assumption for some species. The use of a pure diffusion process with constant diffusion coefficients is based on the assumption that individual insects follow a pure random walk. This assumption is realistic only if individual insects have no long-term memory or large-scale information on the landscape. If individual insects exhibit spatial working memory, their movements cannot be described by a pure random walk and adjustments have to be made to the diffusion model (Ovaskainen, 2008). A more realistic model, from a biological point of view, may include spatial heterogeneity not only in the growth and mortality parameters of the species, but also in the movement patterns of the species. A variation in the diffusion coefficients according to the habitat type may also be assumed if the species exhibits a hierarchical choice with respect to agricultural crop variety and age. In the case of variable diffusion coefficients, the optimal field arrangements may be different from those observed in this study. Further consideration may also be given to the joint effect of more than one pest species and/or diseases on heterogeneous domains similar to those described in this paper. Acknowledgements The financial assistance of the South African Sugarcane Research Institute (SASRI) and the South African National Research Foundation (NRF) towards this research is hereby acknowledged. 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