Population dynamics

We illustrate the analysis of movement using a BQ model with 256 blood feeding sites (denoted by B) and 289 egg laying sites (denoted by Q) that were randomly uniformly distributed. We also developed a BQS model with the same blood feeding and egg laying sites and 225 sugar feeding sites (denoted by S). Each one of the aquatic habitats was identical in every way, yet at the steady state, the population densities at the sites were heterogeneous (Fig 2). Since there were no other differences, the heterogeneity was caused by dispersal patterns.

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Fig 2. The population densities at the steady state for a) the BQ model and b) the BQS model, given the same resource distribution across the landscape.

The red squares are blood feeding sites, the size is scaled to steady state population densities. The blue circles are aquatic habitats , scaled to steady state population densities. The green triangles are sugar sources , scaled to steady state population densities. Aquatic dynamics are identical in every site, so all the variability in population density is due to the position in a network.

https://doi.org/10.1371/journal.pcsy.0000108.g002

Dispersal

In micro-simulation models, resources are located at points in space. Three resource point sets were defined: haunts where mosquitoes rest and where a mosquito might find and feed on a vertebrate host, a point set denoted with elements where describes the location of the element; aquatic habitats where mosquitoes lay eggs, a point set denoted with elements where describes the location of the element; and possibly sugar sources where mosquitoes sugar feed a point set denoted with elements where denotes the location of the element.

Movement among point sets is modeled using matrices that describe where mosquitoes move in a single flight bout, once each time step. The proportion moving from a point in one set to another, from each element in to each element in , is described by a matrix or equivalently . Similarly, the proportion moving from a point in one set to another point in the same set is or (the diagonal describes the probability of staying). The matrices describe the destinations for surviving mosquitoes, so each column sums to one. Mortality associated with dispersing is associated with the source points, and included in .

At the level of a flight bout, mosquito dispersal among point sets is defined by the matrix For mosquito populations, dispersal to blood feed, lay eggs, or sugar feed can be understood through analysis of these population dynamic models, where a single task may require multiple attempts and incur multiple hazards. We have thus developed algorithms that compute the fraction surviving and dispersing through a single phase of the feeding cycle, or through one complete feeding cycle.

Movement to complete a task

Movement to accomplish a task could involve a sequence of search bouts, some of which may be unsuccessful. While we formulate the models with matrices describing movement in a single bout, a functionally relevant measure of dispersal tracks movement through one part of the feeding cycle (Figs 3A and 3B): starting after laying eggs through some failures until a surviving mosquito successfully blood feeds, ; or starting after blood feeding through failures until a mosquito successfully lays eggs .

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Fig 3. Movement to lay eggs or blood feed (top row) creates heterogeneity and structures movement among aquatic habitats or blood feeding sites (bottom row). top row) For one particular model, after defining the movement matrices, , and all the other parameters, we computed (starting from points and dispersing to points to find blood); and (vice versa, to lay eggs). bottom row) From and we computed matrices that show how mosquitoes move through one feeding cycle: (from blood meal to blood meal among points in ); and (from egg batch to egg batch among points in ).

In this visualization, the width of each edge is proportional to the propensity to move, and weaker connections are not plotted. The size of the points is proportional to mosquitoes arriving at each point. Asymmetric flows among pairs of points are represented using arrows.

https://doi.org/10.1371/journal.pcsy.0000108.g003

Since there could be several failures, we modified the models to simulate one part of the feeding cycle. To do so, we modify the equations by setting parameters to zero that determine the fraction of mosquitoes that would otherwise leave the state. We then initialize and follow a cohort, iterating until almost surviving mosquitoes are trapped in the end state (i.e., up to a predefined tolerance). We let denote a matrix describing net dispersal to blood feed once: it is the proportion of mosquitoes leaving after laying eggs that survive and eventually blood feed successfully at each point in . Similarly, we let denote a matrix that describes net dispersal to lay eggs after blood feeding.

The matrices generated by the algorithm describe movement through blood feeding and egg laying, in models with or without sugar feeding. With sugar feeding, a mosquito that has laid eggs must blood feed before laying again. A mosquito may fail to lay eggs and switch to sugar feeding, and then blood feed again before successfully laying eggs (Fig 1). We can compute the matrices and accounting for the fact that a mosquito may sugar feed – possibly multiple times – before ending in the other state. Under the assumptions of this model, there may be some gonotrophic disassociation; a mosquito will always blood feed after sugar feeding, even if it was searching for an aquatic habitat before the transition (Fig 1). In such cases, is the interval between blood meals with or without an egg batch, but because of the delay for parasite development, we must take even greater care in quantifying parasite dispersal by mosquitoes (see below).

Movement through a feeding cycle

After computing dispersal through one part of the feeding cycle, the matrices and define a new, bipartite graph that describes movement among the point sets where mosquitoes lay eggs and where they blood feed (see Figs 3C and 3D). Movement through egg-laying and (possibly) sugar feeding can be represented in block form:

(1)

After blood feeding and egg laying once – one complete feeding cycle – the bipartite graph disconnects into two complementary graphs on their respective point sets:

(2)

The matrix describes movement from haunts to haunts, and describes movement from habitats to habitats (Figs 3C and 3D).

Egg laying

Next, we developed algorithms to compute net dispersal of eggs away from a natal site (see Fig 4). Net lifetime dispersal of eggs from natal aquatic habitats is summed over all gonotrophic cycles, i, until a mosquito has died. If mosquitoes lay o eggs per batch, the expected number of eggs laid per emerging adult female at each site can be computed as:

(3)

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Fig 4. For the same model used in Figs 2A and 3, these plots show lifetime egg dispersal by mosquito populations (), and potential parasite dispersal by mosquito populations ().

The size of each point (dark blue, top) and (dark red, bottom) is proportional to the number of infectious bites or eggs laid arriving at each point (i.e., not arising from). A key feature of these graphs is that some haunts or habitats tend to receive more infective bites or more eggs because of topology, i.e., their position in the graph describing movement. In this visualization, the width of each edge is proportional to the propensity to move, and weak connections are not plotted.

https://doi.org/10.1371/journal.pcsy.0000108.g004

The quantity G is mosquito lifetime egg dispersal kernel for a cohort. We also define net egg dispersal by the population at its steady state

(4)

where is the steady state density of adult female mosquitoes at .

Pathogen dispersal

We developed algorithms to compute potential dispersal of pathogens from the points where mosquitoes blood feed (Fig 4). A basic formula for the dispersion of infective bites away from an infectious host must consider a delay for the pathogen development and a latent period, traditionally called the extrinsic incubation period (EIP). Since chronological and reproductive age are not necessarily in sync, and since there might be gonotrophic disassociation, we can’t use . Instead, we simply run the simulation, summing up all potential transmission events, just as we would compute vectorial capacity.

Net potential dispersal of pathogens is computed by setting an initial cohort of mosquitoes that successfully took a blood meal, B₀, among haunts and iterating the model until all of the mosquitoes are dead. The number of infective bites given by the cohort is found by adding up all the bites after days, where is a parameter describing the EIP. Given B₀, we use the model to compute for a cohort of mosquitoes, and we define V as

(5)

While V defines parasite dispersal by a cohort of mosquitoes among haunts, where each point counts equally, we can also weight parasite dispersal by the steady state population density at each point, which we call potential pathogen dispersion,

where is the steady state density of adult female mosquitoes at .

The matrix defines a quantity that differs slightly from the definition of vectorial capacity, in that it is the expected number of bites arising from all the mosquitoes blood feeding at a haunt, rather than on a single person, computed as if every available host was perfectly infectious. The quantity would be identical to vectorial capacity if it were normalized by the number of hosts available (i.e., modifying B₀), such that the quantity was an expected number arising from a single host. Note also that V and an be used to computes bites arising from or arriving at each haunt by summing over rows or columns.

Dispersal distances

Any pair of matrices describing dispersal propensities among points in a set and the distances among points in that set can be normalized to define a probability distribution function. Since the number of destinations is finite, dispersal is described using probability mass functions (PMFs).

Population structure

The matrices describing mosquito movement among point sets can be understood through the study of the associated graphs: resources are represented by vertices, and movement propensities are represented as directed, weighted edges with a self-loop. We investigated community structure by converting dispersal matrices into graphs and analyzing the results using methods developed for networks. Since most algorithms are designed to analyze simple graphs, we used the walktrap algorithm from iGraph, which can do network analysis for directed, weighted graphs. (Other community detection algorithms could be used, but since walktrap analyzes clustering based on short random walks, it seems to be a good fit for analyzing flights by short-lived mosquitoes.) The algorithm returns a set of communities, groups of points that are more densely connected to each other than to the rest of the points. Information about location is used to develop the dispersal matrices, and location is also used to plot and visualize the results, including the communities identified from analysis of the graphs.

Initially, we sought to analyze the community structure for crude dispersal matrices (for the BQ model, see Eq. 13). The comparison with our other analyses were often at odds. The main difference is that since the mosquito populations are structured by the feeding cycle, a sequence of mosquito flight bouts is not switching randomly among states, but structured by the feeding cycle, from one task to another (Eq. 1). Movement through the feeding cycle is structured by bipartite graphs, and after a feeding cycle, the bipartite graph separated into two independent graphs (Eq. 2) – one for egg laying among aquatic habitats, and another for blood feeding among haunts. Community analysis for crude dispersal matrices (e.g., Eq. 13) did not agree with the analysis for the K² matrices (Eq. 2), in part, because mosquito biology imposes some order to the kind of random walks a mosquito takes.

This discrepancy suggests that a crude analysis of observed mosquito movement patterns might not reveal community structure unless it could somehow account for mosquito behavioral states.

Communities

Our first point sets were generated using randomly uniformly distributed parameters, yet the resulting graphs for describing movement for these basic point sets displayed a high level of organization (see Fig 4). The co-distributions of most pairs of point sets will have some degree of clustering. In these models, the patterns get amplified by the movement rules. Mosquito populations were self-organized into communities around places that had all the required resources nearby. Not surprisingly, the community structures for eggs laid and parasites dispersed tended to overlap geographically (Fig 5). By studying the flows on these bipartite graphs, we began to understand what generated patterns in the steady state densities of mosquitoes at haunts and habitats.

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Fig 5. For the dispersal graphs in Fig 4, we have plotted a subgraph for each community that was identified by the walktrap algorithm (from iGraph) for egg laying (left, ) and potential transmission (right, ).

Each community sub-graph is given a different color After aggregating flows within and among communities, the arrows show the net flows within each patch, plotted as a convex hull around the community. The convex hulls from the complementary process are also plotted in a lighter color in dashed lines. The number at the center is the percentage of the total that remains in each patch. The bent edges show the proportion dispersing out from each patch to every other patch.

https://doi.org/10.1371/journal.pcsy.0000108.g005

These communities could serve as spatial patches in a patch-based meta-population model. Even though we started from a different mathematical formalism, the assumption commonly made for patch-based models – that a constant fraction of mosquitoes in one patch move to other patches – would approximate the processes reasonably well. Notably, any set of patches would approximate the underlying process to some degree: there would be more movement occurring within vs. among communities would also work for any tessellation of the plane. The critical difference is that the meta-population dynamics would approximate the underlying processes better if the patches reflected community structure.

Flows

In analyzing dispersal through a feeding cycle, we noted that the heterogeneous distribution of resources created a propensity for mosquito populations to flow among point sets. The mechanism generating the flow was related to asymmetries in the relative position of points; after leaving a resource at one point in space, finding another resource, and launching a new search from there, the propensity to return to a focal point depended on the arrangement of all the other points nearby. The use of a resource of one type depended on its relative proximity to resources of the other types. For example, in the BQ model, the points that are most likely to be found starting from are not always the same as the points in that are most likely to be found from each starting point . Given the opportunistic nature of searching, mosquitoes would tend to stay in areas where all the required resources can be found, and mosquitoes would tend to leave the neighborhood of resources in one set that are not comparatively close to other resources in the other. These asymmetries in movement propensities can be visualized in the graphs of and (Fig 3). Even in point sets generated using uniformly distributed random numbers, these relationships are rarely symmetrical. The arrangement of points robustly introduces asymmetries that create a propensity for mosquito movement graphs to flow away from some areas and into others.

In analyzing the community structure (see below), we had concerns about the nature of the asymmetric flows and the validity of the analysis of community structure. It is possible, for example, that the net flows in some of these graphs would be so strong and important that they could undermine the cohesiveness of any communities that could form. A graph of turbulent flows of water in a river, for example, could identify important eddies that reflected a kind of transient structure that was constantly broken down by a downstream flow.

After trying several different ways of describing or quantifying patterns, we found the greatest insights from comparing the mosquitoes arriving at each point after a single feeding cycle (i.e., from and ) to population densities at the steady state. The two distributions were similar, which was not surprising, because in these models, population density at the steady state was higher because more mosquitoes moved there. The patterns that emerge in populations reinforce the patterns generated during a feeding cycle. Haunts or habitats with low population densities were found at the periphery of a community, while those with high densities formed the core. Perhaps more relevantly, we did not find any features of the flows – something like the directional flow resembling a river – that would give rise to concerns about the soundness of the network analysis. Instead, we found that the asymmetric flows appear to create and reinforce community structure.

Context and pattern

In developing methods for analyzing mosquito movement on complex resource landscapes, we chose a single landscape where the resources were randomly, uniformly distributed. The analysis of movement in Figs 2–5 all apply to this landscape. The kind of structure that emerged is not a feature of most diffusion-based approaches to understanding mosquito dispersal (but see [13]). This natural patchiness could, perhaps, be handled just as well in metapopulation models where mosquitoes emigrate from a patch at a higher rates when a required resource is unavailable [1]. The framework points out the need for more work to understand the structural limitations of the models we use to describe mosquito ecology as complex systems, but that is beyond the scope of this study.

These behavioral state microsimulation equations are one framework for developing realistic agent-based models of mosquito populations that are easier to analyze than individual-based simulation models [15]. In this framework, mosquito dispersal has a purpose: mosquitoes fly around searching for resources. The movement patterns of mosquito populations are thus structured by those resources. The patterns emerge from searching for different resources in a sequence that follows a mosquitoes behavioral programming. The framework is realistic enough that it might be possible to rigorously define the concept of a mosquito’s fundamental niche as an emergent feature of 1) a mosquito’s behavioral algorithms; 2) the quality, stability, and suitability of aquatic habitats; 3) the availability of other resources; and 4) the costs of searching.

Any framework that is capable of defining a mosquito’s fundamental niche would also describe resource landscapes where mosquitoes would not persist. With enough imagination, patterns of all sorts could emerge. Any competent mosquito ecologist could use these models to understand how context shapes mosquito population dynamics, but the approach would require development of new approaches to studying mosquito populations, perhaps one that is more strongly grounded in evolutionary ecology.

While mosquito population structure can form even if the underlying distribution of resources were drawn from a uniform random distribution, the claim is that mosquito population structure emerges because searching mosquitoes to leave an area while searching for a resource that is not available. To test this, we simulated mosquito populations on a modified version of our landscape by restricting the distribution of one or more resource.

If the population persists, mosquitoes will tend to be aggregated around resources that are rare. To illustrate, we created new landscapes, derived from the original, where one of the resources was rare (Fig 6). These correspond roughly to a rural focus, where aquatic habitats are common but humans are rare (Fig 6A); an urban focus, where humans are abundant, but aquatic habitats are rare (Fig 6B); and finally, a model that demonstrates how the distribution of sugar could explain adult mosquito population distributions, even when human blood hosts and aquatic habitats are common (Fig 6C).

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Fig 6. A set of simulations in which each point set in Fig 2B is replaced by a point set with only 13 points.

The effects on population dynamics are different in each case. Three of these points are reasonably close to each other, and the proximity seems to amplify the local population dynamics. Note that mosquito populations cluster around the point set with least common resource. Rare habitats have the strongest effects on mosquito population structure. Since sugar is an optional resource – mosquitoes can blood feed opportunistically – the effect of sugar being rare is not as strongly limiting.

https://doi.org/10.1371/journal.pcsy.0000108.g006

Behavioral state dynamics on point sets

We define micro-simulation models on point sets representing the locations of resources: haunts where mosquitoes rest and where a mosquito might find and feed on a vertebrate host; aquatic habitats where mosquitoes lay eggs (a point set denoted with elements where ); and possibly sugar sources where mosquitoes sugar feed (a point set denoted with elements where ). In the simulation models, the number of adult mosquitoes emerging from each aquatic habitat on each day is a vector denoted that can be either passed as a parameter or simulated using a population dynamic model for aquatic immature population dynamics (see below). Adult mosquitoes are located at one of these points at each point in time (e.g., one day), represented as a set of vectors: , the number seeking blood at haunts; , the number attempting to lay eggs at habitats; or , attempting to feed at sugar sources. Mosquito bionomics can depend on both behavioral state and location, including daily survival (vectors denoted or or ), and daily blood feeding or egg laying success ( or or ). For convenience in writing the equations, below, we use a symbol to denote the complement: .

Movement among point sets is modeled using matrices that describe where mosquitoes move each time step. The proportion moving from a point in one set to a another, from to , is described by a matrix or equivalently . Similarly, the proportion of mosquitoes that are re-attempting a task that move from a point in one set to a point in the other is . The matrices describe where surviving mosquitoes end up; all mortality is described in the survival parameters (p). Since there are a finite number of destinations, each column is a probability mass function (PMF).

Immature population dynamics

For purposes of integrating the effects of movement and aquatic ecology, we assume each female lays o eggs, so the total number of eggs laid each day in each site is:

(6)

We assume the number of immature mosquitoes in aquatic habitats, denoted , is subject to density dependence, which could delay maturation or increase mortality. The maturating fraction is, where , and surviving fraction is where . The parameters are site-specific, so that some habitats can vary in quality. The dynamics are:

(7)

The number of adult females emerging each day is half the mosquitoes who both survived and matured:

(8)

While this model is adequate for the needs of this study, the model may not robustly capture some relevant features of mosquito ecology or larval source management, such as when larval population structure and delays are important. The software, ramp.micro, includes alternative models with stage structure.

A basic feeding cycle model

In the basic feeding cycle model over one time step, mosquitoes either attempt to blood feed or attempt to lay eggs. The result of an attempt is either survival or death, and if the mosquito survives, success or failure. A success moves a mosquito to the other state, and if they fail, they must try again. Either way, the mosquito moves to a point in the set for the resource they seek: the diagonal of is the probability of staying. Using instead of for the diagonal matrix, the equations are:

(9)

where is the number of emergent adult female mosquitoes on day t (see above).

A feeding cycle model with sugar feeding

In the models with sugar feeding, the updating basic rules are similar, but a new set of parameters are required to describe the frequency of switching to sugar feeding from other states. Each day, some fraction of mosquitoes switch to a sugar feeding state from various points in the feeding cycle (Fig 1): the switch to sugar feeding occurs in a fraction of recently emerged mosquitoes ; after egg laying ; after a failed egg laying attempt, ; or after a failed flood feeding attempt, . We assume that all mosquitoes revert to blood feeding after sugar feeding, implying that the mosquito resorbed the eggs. We also assume that mosquitoes would never attempt to sugar feed after a blood meal, but that they would instead always attempt to lay eggs at least once (Fig 1).

(10)

Resource landscapes and dispersal matrices

To simulate mosquito dynamics, it is necessary to fully specify all its elements, including the parameters and resource point sets. Dynamics and patterns emerge from the co-distribution of those points and from the population processes. Aggregate movement patterns of mosquito populations in models of this type are determined by how many sites exist and which sites a mosquito ends up finding. Patterns emerge as movement and survival are iterated over multiple search bouts. To develop some analytical methods and some basic intuition, we start with randomly generated sets of points and simple distance-based movement rules.

We generated sets of points within a spatial domain using various probability distribution functions, including a uniform random distribution, lattices, and clustered points generated by various algorithms.

We use functions to generate the probability of finding a point as a function of its distance and context. Let denote the distance from point i to j. We define some function to assign a weight to every pair of points. In this model, we use the exponential family of functions to assign search weights:

(11)

The scalar is a linear weight on each destination making it more or less attractive from every distance. In ramp.micro, other functional forms have been developed, including shapes that have a fat tail.

The search weights are normalized and translated into proportions: the proportion arriving at each point in a destination set index by j from a source set indexed by i is normalized across all destinations from each starting point:

(12)

To be clear about the assumptions of this model, a matrix, , describes the probability of arriving at a destination conditioned on the mosquito surviving the flight, so the columns of sum up to one. Mortality associated with dispersal away from a point is included in the parameter describing daily survival of a mosquito leaving from each point. Each column of each thus defines a PMF for surviving mosquitoes, but statistics describing movement of mosquitoes depend on how many mosquitoes in the population start from each point and the resource they seek.

In the models we examine herein, the dispersal matrix is not time-dependent, but time-dependent matrices could be adopted to simulate other dispersal rules and effects, such as wind with variable speed and direction.

Simulation

For any landscape, we can simulate mosquito population dynamics over time. If all parameters are constant over time (i.e., values of p, , , , , and ), then we can find a steady state either by inverting a matrix or by running the simulation for a long time. We let , , and denote the steady state densities of adult mosquitoes in the point sets.

Community detection

We analyzed all of the square community matrices defined on single point sets, including , , G, V, , and .

We were also interested in analyzing the matrix describing the bulk movement of adult, female mosquitoes among all patches. For the model without sugar feeding, movement propensities by mosquitoes through one flight bout are described by the matrix:

(13)

We also analyzed the closely related matrix that describes the movement propensities of mosquito populations, which is weighted by steady state densities:

(14)

The matrix for the graph with sugar feeding is to Eq. 10 as Eq. 14 is to Eq. 9.

For any movement matrix, we examined population structure by analyzing these matrices as graphs. To summarize movement patterns, we identified “communities” defined by these graphs. A community expects more transitions (mosquito jumps) amongst the points within the community than transitions to a different community. Since the graphs are potentially asymmetric, we used the walktrap algorithm from the R package iGraph.

Notably, mosquito survival and dispersal work like the walktrap algorithm for community detection. The short lifespan of mosquitoes means that most mosquitoes will tend to move approximately times in random walks, where is something like a geometric mean lifetime probability of survival.

The graphs we analyzed were directed, weighted graphs with a self-loop. To understand these graphs, we decomposed them into two related graphs: the matrix D describing a graph can be expressed as the sum of a symmetric matrix and an asymmetric matrix that represents the residual flows . The symmetric matrix is the minimum of the matrix and its transpose, :

(15)

and the residual flow is:

(16)

Note that for all that . If , then for all pairs i,j, either or .

We examined the residual flows created by and separately to better understand the nature of the mosquito population movement patterns that emerge from these simple rules.

To examine the robustness of the community structure, we run walktrap both on the graphs and on their decomposed matrices (i.e., for each graph D that we examined, we also examined and ). For an extended discussion, see the articles at https://dd-harp.github.io/ramp.micro/.