ALMaSS (the Animal Landscape and Man Simulation System) is a well-established modelling system known for its detailed agent-based animal models. However, for certain species, such as aphids, extremely high population sizes and limited detailed individual behaviour information make agent-based approaches either computationally expensive or unfeasible. Additionally, when these species are modelled as a prey base for other species, such as aphids for ladybirds, an efficient and accurate modelling approach is required. To address these challenges, we introduce a subpopulation-based model framework that effectively handles high population densities, while reducing computational complexity. This framework divides the landscape into a non-overlapping grid and population dynamics are managed within each grid using a flexible stage-structured population model. Grids are interconnected through short- and long-distance dispersal mechanisms, allowing for realistic movement and interactions. Parallel programming is employed to enhance performance, enabling faster simulation runs. We demonstrate the application of this model framework with a hypothetical species (not a species in the real world) and showcase its use in modelling aphid populations. This foundational subpopulation model could be used to model species with large population sizes within the ALMaSS framework and can be integrated into other simulation systems.

ALMaSS, dispersal, landscape simulation, mortality, reproduction, subpopulation modelling

The Animal Landscape and Man Simulation System (ALMaSS) (Topping et al. 2003) is a simulation system of agent-based models developed over the past 20 years for evaluating the impact of management on organisms simulated in realistic landscapes. ALMaSS is focused on agricultural systems issues and has been used particularly to evaluate agri-environmental scheme implementation (e.g., Langhammer et al. 2017; Topping et al. 2019) and impact assessment of pesticides (e.g., Topping et al. 2018; Ziółkowska et al. 2021). To evaluate management impacts using ALMaSS, the species models are integrated into a highly detailed landscape simulation (Topping and Duan 2024a). ALMaSS landscape simulation has a resolution of the underlying topographical map of 1 m² and simulates daily farm and field management and plant growth.

Until now, all ALMaSS species models have been developed as agent-based models (Duan et al. 2022; Capela et al. 2024; Topping and Duan 2024b) simulating detailed individual behaviour and decision-making (e.g. Topping and Odderskaer 2004; Topping et al. 2010). Agent-based models use the local informational context and the model species’ own condition to make decisions to further an agenda of development, movement, survival and reproduction. The result is an emergent population response, based on the action and interaction between these agents and between the agents and the environment. However, the agent-based approach is not applicable for certain species (e.g. aphids) due to a lack of detailed individual-based information, the huge numbers of individuals that can be widespread across a landscape or both. In addition, these species are often the prey for other species, which could be modelled as agents (e.g. aphids as prey for ladybirds). Thus, we need to model billions of individuals time efficiently since they need to be run simultaneously with other agent-based models. Hence, we had to develop a new ALMaSS population management approach to include these species in the simulation system.

The key design objectives for the new approach were as follows:

• Be capable of representing the entire population for the simulated landscape;
• Include the detailed landscape representation and be able to interact with landscape, field and crop management;
• Include spatial dynamics of the species;
• Accurately represent the population development of the species;
• Run efficiently on landscapes of standard 10 × 10 km.

The proposed subpopulation model integrates three types of model: stage-structured population models, spatially-structured population models and landscape simulation. Previous models typically use one of these approaches. The Leslie Matrix is one of the best-known stage-structured population models (Leslie 1945). While the Leslie Matrix model specifically focuses on age-structured populations, stage-structured models encompass a broader range of characteristics, including life stages that may not be directly tied to chronological age. In stage-structured population models, individuals are classified into different stages, based on specific characteristics, such as size, developmental stage or reproductive status. Each stage is associated with its own set of transition probabilities, representing the likelihood of individuals moving from one stage to another. This type of model is particularly useful when age alone does not fully capture the variations in the life history of a species. Using this form of model as a base allows a large degree of flexibility in defining our subpopulation model methods.

Bommarco et al. (2007) used a lattice model and individual dispersal of carabids to simulate an aphid/carabid system and Parry et al. (2006) used a detailed individual-based model to simulate aphids in relatively simple landscapes. The closest model to the proposed model in concept is probably the DYMEX model, which also links a more traditional population approach to spatial dynamics through dispersal (Parry et al. 2011). Other approaches to modelling populations, such as aphids, include the use of a more traditional mathematical simulation approach, in some cases based on spatially distributed observations (e.g. Knudsen and Schotzko 1999; Winder et al. 2001; Dutta et al. 2008). Similarly, Damgaard et al. (2020) used a stochastic aphid population growth model and fitted the model to an empirical spatial time series of aphid population data using a Bayesian hierarchical fitting procedure to create predictions of aphid population in time and space. In the case of Brown et al. (2021), a multi-patch model was developed to look at the dynamics of aphids and ladybird predators, using a differential equation approach expanded to include details of the attraction of the habitats and details of dispersal.

Our approach adapts a standard stage-structured population model to create a spatially dynamic model interacting with ALMaSS landscape models. We believe the combination created here to be unique because of the integration with the detailed landscape modelling. The stage-structured population model by Lefkovitch (1965) described population growth by breaking the population down into stages (often physiological life stages with an individual life history (Botsford et al. 2019)) and tracking the development of the population by describing rates of change from one stage to another. We used this basic concept for the ALMaSS subpopulation model to create a general subpopulation model that can be easily adapted to represent different species with different life cycles and types of behaviour. To implement the spatial dynamics, we split the entire population in the simulated landscape into a grid of stage-structured subpopulations, each potentially linked to all others by dispersal. For each life stage, the abundance at different ages is tracked daily for every grid. Within the subpopulation, the standard population processes of reproduction, mortality, development and dispersal are represented by separate, but linked modules within the model. The model can be tuned to represent a wide range of species by specifying the values of parameters for vital rates and altering the structure of the table describing the stage-structured population.

This paper describes and documents the foundational ALMaSS subpopulation model framework, specifically designed to model species with large population sizes. It is important to note that the collection of biological information and parameterisation for specific species are beyond the scope of this paper. Instead, we demonstrate the model’s design objectives and general applicability through a hypothetical generic species and provide examples of aphid models developed using this subpopulation model framework.

The implementation of the general subpopulation model within ALMaSS is achieved through two C++ classes, “Subpopulation” and “Subpopulation_Population_Manager”. The “Subpopulation” class represents a single subpopulation within a grid cell and an object of this class is used to simulate the stage-structured subpopulation in each grid cell (with a default size of 10 m × 10 m). The “Subpopulation_Population_Manager” class is an administrative class that manages the entire population across the simulated landscape, which comprises all “Subpopulation” objects and provides input/output functions for the subpopulation model.

These two classes contain all the necessary code to implement the subpopulation model framework, which can be based on modelling specific species in the real world, such as aphids (Thomsen et al. 2024). The proposed model is integrated with the standard ALMaSS landscape simulation model, which provides the model species with dynamic, landscape-specific information as the simulation progresses. This integration allows the species to interact realistically with its environment throughout the simulation.

This paper’s description of the general subpopulation model assumes a hypothetical generic species (not based on any real-world species, purely for illustration purposes) with five life stages: egg, larva, pupa, winged adult and wingless adult. Adults are only females and they are all mated and are capable of laying eggs. The winged adults are the only life stage capable of long-distance dispersal through flight, which they can undertake once in their lifetime. Since their flight is influenced by wind direction and speed, it is considered partly passive transportation. After flying, they become wingless adults. Local movement occurs by shifting a certain proportion of the subpopulation to one of the adjacent cells surrounding the current location. Thus, the distance they can cover in a single short-distance dispersal event depends on the size or dimensions of these cells. Eggs, larvae and pupae are immobile within their grid cells. When applying the general subpopulation model to a specific species, biological details such as life stages (as shown in Table 1) must be incorporated into the proposed model to accurately reflect the species’ characteristics.

The Subpopulation_Population_Manager class

This class represents the entire population across the simulated landscape as a collection of subpopulations. For simplicity, the “Subpopulation_Population_Manager” class will be referred to as the manager class throughout the rest of this paper. Subpopulations are defined by applying a regular grid (default size of 10 m × 10 m) over the simulated landscape. For each grid cell, an object of the “Subpopulation” class is created and managed daily by the manager class. For example, a 10 km × 10 km landscape with a 10 m × 10 m grid size would result in 1 million “Subpopulation” objects. Fig. 1 illustrates the running flow of the manager class, showing initialisation and the daily time-step loop, which continues until the simulation concludes.

Figure 1. 

Running flow of the Subpopulation_Population_Manager class.

When an instance of the manager class is created, initialisation is triggered. During this process, the manager loads all parameters and host information (see Table 1) and divides the landscape into grid cells. Each grid cell is assigned an instance of the “Subpopulation” class, with the suitability of each cell for the species calculated, based on the landscape composition within that cell. A key procedure during initialisation is the pre-calculation of the long-distance dispersal landing mask (see “Long-distance dispersal pre-calculation”). After initialisation, the manager enters the main simulation loop, advancing in one-day time steps and repeating until the simulation is complete.

DoFirst function

The first part of the main simulation loop is to call the DoFirst function of the manager class. Here, the development season is updated first. Some species, for example, black bean aphids, switch plant hosts on which they have different life cycles. Based on this host switch, the development season is defined. If the model species do not have this behaviour, the development season will not change. Afterwards, the common part of the mortality rate depending on the environmental factors (e.g. temperature), which is identical for the whole simulated landscape at each time step, is also calculated. The flying animals will be distributed using the landing mask, based on the wind speed and wind direction for that day in this DoFirst function as well.

BeginStep, Step, EndStep functions

After the preparatory steps in the DoFirst function, a sequence of three loops for all the “Subpopulation” objects is initiated to call functions from the “Subpopulation” object. The first loop calls the BeginStep function for all “Subpopulation” objects. Although this function is typically empty in the base class, it can be overridden in a derived model class if needed. The purpose of BeginStep in a derived class is to prepare the “Subpopulation” objects for the next loop, where their Step function is called.

The second loop runs immediately after the BeginStep loop, iterating over all “Subpopulation” objects to execute their Step function. The Step function encompasses behaviour related to population density, habitat suitability, individual development, reproduction, mortality and dispersal. Once the Step function has been executed for all “Subpopulation” objects, the third loop calls the EndStep function for each one to conclude the day’s activities. Like BeginStep, the EndStep function is empty in the general subpopulation class, but can be overridden in a derived class if specific actions are required at the end of the time step for the real model species.

These three loops are parallelised using OpenMP (Chandra 2001) at the thread level, which significantly enhances performance. Given the potentially large number of grids (up to 1 million in standard configurations), processing grids sequentially would be extremely time-consuming. Parallelisation is, therefore, a key design feature that enables the model to run efficiently, even with a large number of “Subpopulation” objects.

DoLast function

After cycling through all “Subpopulation” objects, common calculations shared by all the “Subpopulation” objects are executed within the DoLast function. These calculations yield landscape-wide results used by all subpopulations. The doDevelopment function calculates the accumulated day degrees, which will be detailed in “Development calculation (doDevelopment function)”. The doSpeciesLastThing function, while empty in this general subpopulation model, offers a placeholder for potential species-specific overrides. To advance simulation time, addNewDay increments to the day counter. Finally, updateWholePopulation and subpopuBaseOutputProbe provide population recording, the former aggregating totals and the latter recording daily numbers.

Other functions

Alongside core calculations, the manager provides four additional functions for “Subpopulation” objects. These are the next life stage calculation, reproduction calculation and functions linked to the development calculation (supplyOldEnoughIndex and supplyAgeInDay) (see Fig. 1). These are described in the relevant sections below.

Long-distance dispersal pre-calculation

This function is used to pre-calculate the landing mask for the model species doing the long-distance dispersal for the sake of computing efficiency.

We assume the long-distance flying dispersal is influenced by the wind, but this can only happen when the wind speed is lower than a certain value. Long-distance flying is done using 2D landing masks, which are generated by the landing proportion 1D curve along the wind direction. Below are the key parameters with the default values in the parenthesis for the long-distance flying dispersal.

• Maximum wind speed, w _M (12 m/s): When the wind speed is larger than w _M , the model species will not attempt to fly;
• Wind step size, w _S (2 m/s): This is used to sample the wind speed to discrete ranges. Each range has four landing 2D masks in four wind directions;
• Peak distance, d _p (100 m): This is the distance where most of the flying ones can land along the wind direction when the wind speed is in the first range (0 to w _S) ;
• Longest distance, d _l (1000 m): This is the longest distance at which the flying ones can land along the wind direction when the wind speed is in the first range (0 − w _S) ;
• Wind speed scale, S _W (1.2): This is the scale used to increase the peak and longest distance when wind speed increases to one range from the previous range.

The landing curve along the wind direction

Two polynomial functions are used to generate a long tail landing curve along the wind direction, as shown in equation (1):

$p\left(l\right)=\left\{\begin{array}{cc}-l^2+2d_{p}l,& \text{if}l\le d_p\\ \frac{d_p^2{\left(d_l-l\right)}^2}{{\left(d_l-d_p\right)}^2},& \text{if}l>d_p\text{and}l\le d_l\end{array}\right.$ (1)

where l is the distance from the landing position to the original position. In this way, the position at d_p will have the largest landing proportion compared to other positions. The dispersers can land at a maximum distance of d_l, beyond which no one can land (p (l) = 0). The landing curves calculated using the default parameter values can be found in Fig. 2. When the wind speed enters a new (higher) wind speed, range, d_p and d_l are multiplied by the wind speed scale (S_W =1.2 by default) for the calculation in equation (1).

Figure 2. 

Landing curve examples calculated using equation (1) for five wind speed ranges. When the speed is larger, the model species can fly further away from their originations (denoted by 0 on the x-axis).

2D landing mask

After obtaining the landing curve along the wind direction, we need to convert it to a 2D mask in order to land the dispersers with wings on a 2D landscape. Firstly, we rotate the landing curve 360 degrees through the original location. If we suppose that the original location is at (0,0), then the distance from each location denoted by (x,y) in a 2D landscape to the original location can be calculated using equation (2):

$l=\sqrt{x^2+y^2}$ (2)

After equation (2), we can obtain the initial symmetric 2D landing mask to the source location using equation (3):

$L^{\text{'}}(x,y)=p\left(l\right)=p\left(\sqrt{x^2+y^2}\right)$ (3)

Next, we skew the landing mask, based on wind direction. We want more individuals to land downwind and none to land directly upwind. To achieve this, we use the cosine of the angle between the dispersal path (origin to destination) and the wind direction as a weight.

We suppose the wind direction to the source location is given by (b, −a) and the direction from the destination location to the source location is given by (y, −x). Then, the weight used to skew the initial landing mask can be calculated using equation (4):

$w_c(x,y)={\left(\frac{by+ax}{\sqrt{x^2+y^2}\sqrt{a^2+b^2}}\right)}^m$ (4)

where m is an integer to control the landing range; when m is larger, more individuals will land along the wind direction. Afterwards, the 2D mask with the wind direction of (b, −a) can be calculated by equation (5) as exmapled in Fig. 3.

Figure 3. 

2D landing mask examples with the colour bar for the landing proportions. The winged adult flies from the original (0, 0). The wind direction is along the y-axis (from bottom to top). The wind adults will not land at the locations where the flying direction is opposite to the wind direction. Further, when the wind speed is increasing, they can reach further locations along the wind direction. m in equation (4) is set to be 3 in these examples.

Four wind directions (N, S, W, E) are used in the proposed model. Each wind speed range has one landing mask for every wind direction. The landing masks are normalised so their sum equals 1, ensuring appropriate distribution during landing calculations. One example of the 2D mask with a wind direction for five wind speed ranges can be found in Fig. 3.

$L(x,y)=\left\{\begin{array}{c}{L}^{\text{'}}(x,y)w_c(x,y),\text{if}{w}_c(x,y)>0\\ 0,\text{else}\end{array}\right.$ (5)

In this paper, we introduced a general subpopulation model framework designed to handle species with large populations. This framework operates by tracking the population development of species across a simulated landscape, recording the number of individuals at various life stages and ages within each grid cell. By aggregating these local subpopulations, the model can represent the entire population across the landscape, providing a comprehensive view of population dynamics.

Our model differs from traditional agent-based models by operating on subpopulations belonging to different non-overlapping grid cells in the landscape rather than individuals, using the number of individuals at specific life stages and at different ages as the minimal unit. While this approach offers less granularity than modelling each animal as an individual agent, it still achieves a fine resolution by allowing stage-specific age distinctions (e.g. all 12-day-old larvae). This balance enables us to capture detailed population dynamics without the computational overhead of full agent-based modelling, making it particularly suited for large-scale simulations.

By integrating with existing ALMaSS landscape models, our subpopulation framework can incorporate dynamic environmental factors, such as vegetation growth and agricultural practices. This connection allows the model species to interact realistically with their landscape, responding to factors like field management decisions (e.g. insecticide treatments) and seasonal vegetation changes. As demonstrated in Fig. 5, aphids modelled using the proposed subpopulation model framework appear only in suitable habitats and migrate into crop areas as they become available, reflecting how the model accounts for both spatial and temporal landscape variations.

Our model links subpopulations in different cells to capture spatial population structure through dispersal mechanisms rather than relying on traditional metapopulation or patch dynamic models. Dispersal mechanisms include long-distance flights and local movement between grids, facilitating accurate spatial dynamics across the landscape. The landscape model provides a dynamic context, affecting subpopulation behaviour through factors like wind speed and direction. This approach allows for the simulation of complex dispersal scenarios, where subpopulations can respond adaptively to changing environmental conditions.

Our multi-threaded approach enables efficient simulations, even for complex scenarios. For instance, we successfully ran a 30-year simulation involving four aphid species and one ladybird species running simultaneously across a 10 km-by-10 km landscape in just 20 hours using 30 threads. This performance demonstrates the model’s capability to handle extensive simulations, while maintaining computational efficiency.

While the model’s conceptual structure is relatively straightforward, its implementation requires a solid understanding of the ALMaSS system and its programming environment, reflecting a trade-off between complexity and flexibility. Despite this, the framework is sufficiently generic to model a wide range of species, making it a versatile tool for future ecological modelling. Provided as a set of C++ classes, this model can be directly integrated into the ALMaSS software environment or adapted to other modelling systems capable of supplying the necessary data. This flexibility makes it a valuable resource for modellers seeking to explore large-scale population dynamics in complex landscapes.

This paper presents a novel base model designed to serve as a foundational framework for modelling species with large populations. This subpopulation model provides a flexible tool for developing species-specific simulations by focusing on population dynamics within detailed simulated landscapes. Our approach enables a more accurate understanding of types of population behaviour without requiring individual-level modelling by integrating elements from stage-structured and spatially structured population models with landscape simulation. Unlike previous models that often rely on a single approach, this framework leverages the strengths of multiple methodologies. It combines detailed landscape modelling with nuanced subpopulation dynamics by segmenting landscapes into non-overlapping grids, allowing for the simulation of large-scale systems involving billions of individuals. Already successfully applied to simulate aphid populations, this model serves as a robust foundation for building new, species-specific models within ALMaSS or other simulation projects. These models can be tailored to meet the unique requirements of different species.

We would like to thank Sara Nafisi for her assistance with the figures in this paper.

The source code and the running directory are publicly available at:

https://gitlab.com/ALMaSS/almass_methodology.git

The code documentation for the subpopulation model is publicly available at:

https://almass.gitlab.io/almass_methodology/Subpopulation.pdf