Coffee agroforestry systems in Central America: II. Development of a simple process-based model and preliminary results

Abstract

Research on coffee agroforestry systems in Central America has identified various environmental factors, management strategies and plant characteristics that affect growth, yield and the impact of the systems on the environment. Much of this literature is not quantitative, and it remains difficult to optimise growing area selection, shade tree use and management. To assist in this optimisation we developed a simple dynamic model of coffee agroforestry systems. The model includes the physiology of vegetative and reproductive growth of coffee plants, and its response to different growing conditions. This is integrated into a plot-scale model of coffee and shade tree growth which includes competition for light, water and nutrients and allows for management treatments such as spacing, thinning, pruning and fertilising. Because of the limited availability of quantitative information, model parameterisation remains fraught with uncertainty, but model behaviour seems consistent with observations. We show examples of how the model can be used to examine trade-offs between increasing coffee and tree productivity, and between maximising productivity and limiting the impact of the system on the environment.

Appendices

Appendix 1. Rainfall generator for Central America

This appendix describes an algorithm for generating time series of daily rainfall from monthly total rainfall. The following figure shows the results of 365 consecutive daily rainfall measurements at Heredia, Costa Rica (10.03°N, 84.14°W, 1180 m altitude), from 1 April 2003 to 31 March 2004:

The figure shows: (a) daily rainfall, (b) the 31-day moving average, (c) the 31-day moving total of rainy days. Average daily rainfall was 5.4 mm with a standard deviation of 9.3 mm, and the total number of rainy days (rain > 0) was 242. Nonlinear regression of (c) on (b) leads to an exponential relation (r² = 0.90) for the fraction of days without rain as a function of monthly total rainfall:

$$ {\text{f}}_{\text{dry}} = {\text{e}}^{{ - 0.00 8 7 \Upsigma {\text{rain}}}} $$

(A1)

where f_dry is the fraction of days without rain in a given period of 31 days (−) and Σrain is total rainfall in the 31 days. The amount of rainfall on rainy days follows a nearly exponential distribution with the following cumulative density function:

$$ {\text{P}}\left( {{\text{rain}} < {\text{x}}} \right) = 1- {\text{e}}^{{ - {\text{x}}/{\text{rain}}_{\text{wet}}}} $$

(A2)

where P is the probability that rainfall is less than x mm on a given rainy day and rain_wet is the average rainfall on a rainy day (mm), calculated as:

$$ {\text{rain}}_{\text{wet}} = \Upsigma {\text{rain}}/\left( { 3 1\left( { 1- {\text{f}}_{\text{dry}} } \right)} \right). $$

(A3)

These equations can be turned into a 5-step algorithm for generating daily rainfall amounts from monthly data: (1) Interpolate the monthly rainfall data to generate a day-by-day time series, (2) Calculate the corresponding time series of f_dry using Eq. A1, (3) For each day, assume it is dry if a randomly chosen number between 0 and 1 is less than f_dry, (4) For the remaining days calculate the expectation value of rain using A3, (5) For each rainy day, estimate the amount of rain as the expectation value times minus the logarithm of a randomly chosen number between 0 and 1.

Because the algorithm is stochastic it does not reproduce any observed rain data exactly. However, over 90% of generated time series for the Heredia site, matched the observed mean and standard deviation of daily rain fall, and the number of rainy days in the year, within 5, 10 and 5%, respectively. The effectiveness of the algorithm was further verified using the rainfall data from the Turrialba site.

Appendix 2. Spatial dynamics: updating variables when the shaded area changes

In agroforestry systems, the fraction of the field shaded by trees increases when the tree crowns expand and decreases when the trees are pruned or thinned. In our model, the spatial dynamics of shade require that the values of the state variables are continuously updated. The state variables in the model are all expressed per unit shaded or unshaded surface area. When area changes from unshaded to shaded, or vice versa, the state variables in the expanding part of the field become “mixed” with the corresponding variables in the other part. Say A ₁ and A ₂ are the sun- and shade-areas, X ₁ and X ₂ are totals of any state variable in these areas, and x ₁ and x ₂ are the values of the state variable per unit area (x _i  = X _i /A _i ). For example, X and x may be coffee biomass (kg) and coffee biomass density (kg m⁻²), respectively. Then, the following statement from calculus applies:

$$ {\text{d}}x_{i} /{\text{d}}t = {\text{d}}\left( {X_{i} /A_{i} } \right)/{\text{d}}t = \left( {A_{i} {\text{d}}X_{i} /{\text{d}}t - X_{i} {\text{d}}A_{i} /{\text{d}}t} \right)/A_{i}^{ 2} $$

If all changes in the x _i are due to changes in shaded area, we can eliminate the X _i and rewrite the statement in terms of x _i and A _i only:

$$ {\text{d}}x_{i} /{\text{d}}t = {\text{Max}}\left[ {0,{\text{d}}A_{i} /{\text{d}}t/A_{i} } \right]*\left( {x_{j} - x_{i} } \right) $$

where x _j is the state variable density in the part of the field complementary to x _i . If the x _i also change because of processes within the two areas (e.g. coffee growth or senescence), we add terms for those processes to the above formula for dx _i /dt.

The above formula thus affords an easy means of keeping track of changes in state variables in any agroforestry model where two types of ground cover are distinguished.