Differential equation model for central-place foragers with memory: implications for bumble bee crop pollination

Abstract

Bumble bees provide valuable pollination services to crops around the world. However, their populations are declining in intensively farmed landscapes. Understanding the dispersal behaviour of these bees is a key step in determining how agricultural landscapes can best be enhanced for bumble bee survival. Here we develop a partial integro-differential equation model to predict the spatial distribution of foraging bumble bees in dynamic heterogeneous landscapes. In our model, the foraging population is divided into two subpopulations, one engaged in an intensive search mode (modeled by diffusion) and the other engaged in an extensive search mode (modeled by advection). Our model considers the effects of resource-dependent switching rates between movement modes, resource depletion, central-place foraging behaviour, and memory. We use our model to investigate how crop pollination services are affected by wildflower enhancements. We find that planting wildflowers such that the crop is located in between the wildflowers and the nest site can benefit crop pollination in two different scenarios. If the bees do not have a strong preference for wildflowers, a small or low density wildflower patch is beneficial. If, on the other hand, the bees strongly prefer the wildflowers, then a large or high density wildflower patch is beneficial. The increase of the crop pollination services in the later scenario is of remarkable magnitude.

Appendices

A Parameters

A.1 $D_H$

$D_H = 0.06 ~\mathrm {km}^2$/h is computed using the method described in Chapter 2 of Okubo and Levin (2013).

A.2 $v_{CPF}$

$v_{CPF}=0.55 ~\mathrm {km/h}$ is the value of the central place foraging velocity that make bees stay in a realistic home range of around 4km (Greenleaf et al. 2007) when the resource in the landscape is relatively abundant and uniform distributed.

A.3 $k_{SHb}$, $k_{SHw}$ and $k_{HS}$

In Table 1 of Woodgate et al. (2016), we have radar tracking data corresponding to the “flight duration” and to the “time in flight” for different bees during multiple days. In our work, we have considered that the time in flight is the time spent scouting whereas the remaining time (“flight duration”-“time in flight”) is the time spent harvesting.

All the data in Woodgate et al. (2016) together gives us a total of 251 h of flight duration and 10.2 h of time in flight. We can now use those times to measure the relative values of $k_{SH}$ (rate for the transition $H\rightarrow S$) and $k_{HS}$ (rate for the transition $S\rightarrow H$):

$$\begin{aligned} k_{HS} = \frac{10.2}{251} = 0.040637, \\ k_{SH} = \frac{251-10.2}{251} = 0.959363. \end{aligned}$$

It is important to note that scouting times here includes both (“memory scouts” and “no memory scouts”).

The parameters $k_{HS}$ and $k_{SH}$ give the expected proportion of scouting and harvesting bees in the equilibrium situation. However, they do not contain information about how fast the equilibrium distribution is reached. From Lihoreau et al. (2012) we can deduce that the equilibrium should be reached in around (2–3) h. By looking at the results of our simulations, we have decided to multiply by a factor 1000 the previous values found for $k_{HS}$ and $k_{SH}$, such that the equilibrium happens after around (2–3) h:

$$\begin{aligned}&k_{HS} = \frac{10.2}{251} = 40\\&k_{SH} = \frac{251-10.2}{251} = 960 \end{aligned}$$

.

In addition, we would also like to distinguish between high quality resources and low quality resources by making the transition from scouting to harvest to be more or less frequent depending on the resource quality. For that reason, we define two flower species with a different scout-harvester switching rates: $k_{SHb}$ and $k_{SHw}$. We decided to keep $k_{SHw}=960$ for one flower type while changing the quality (type) for the other; $k_{SHb}=(960,240,20)$.

A.4 $\lambda$ and $\beta$

In Table 1 of Woodgate et al. (2016), we have radar tracking data informing about the “total flights” and the “total number of exploitation flights”. Aligned with the analogy we have made about explorers being no-memory scouting bees and exploiters the memory scouting bees, we can now use the data in Woodgate et al. (2016) to compute the fraction of memory scouts with respect to the no-memory scouts.

All the data together gives a total of 244 flights of which 182 were exploitation flights. We can now use this data to compute $\beta$ (proportion $S_n/S$) and $\lambda$ (proportion $S/S_n$):

$$\begin{aligned}&\lambda = \frac{182}{244} = 0.756 \\&\beta = \frac{244-182}{244} = 0.254. \end{aligned}$$

A.5 $\eta$ and $\kappa$

In Lihoreau et al. (2012) (similar results are in Woodgate et al. (2017)) it is said that Bumble bees stabilize its foraging route after around 30 foraging bouts. Taking into account that the mean flight duration in those studies was of around 4.5 min, we can deduce that the memory function should stabilize after around: $30\cdot (4.5) =$ 2.5 h.

In Lihoreau et al. (2012) it is also said that after removal of a flower patch, bees keep visiting the past patch for around 8 long foraging bouts, were the flight duration of each bout is of around 20 min. From this data we can deduce that the memory function should decay after a time of around: $8\cdot 20 \approx$ (2–3) h.

After running simulations in different landscape scenarios, we found that the values of $\kappa$ and $\eta$ from Eq. (2.5) that better fit the mentioned data are:

$$\begin{aligned} \eta = 500 \qquad \kappa = 20 ~\mathrm {h}^{-1}. \end{aligned}$$

A.6 $\sigma$

In Eq. 2.5 we say that memory direct scouting bees towards those locations with high density of harvesters. We bounded the “attraction” with the distance in such a way that closer harvesters are more “attractive” than those far away. The distance dependence of the “attraction” is chosen to be a Gaussian distribution centered at distance zero and with the deviation being of the order of magnitude of the home-range of Bumble bees (Greenleaf et al. 2007), that is:

$$\begin{aligned} \sigma = 4~\mathrm {km} \end{aligned}$$

A.7 $D_{Sn}$

The fast diffusion undertaken by no-memory scouting bees helps them in their role of exploring the landscape for new resources. The value of the diffusive constant is related to the learning time of new patch resources in the landscape. From Lihoreau et al. (2012) we can deduced that the learning occurs in about (2–3) h. We have run simulations and check that the value of $D_{Sn}$ that makes the learning to happen in approximately (2–3) h is:

$$\begin{aligned} D_{Sn} = 1.2~\mathrm {km}^2 \mathrm {/h}. \end{aligned}$$

A.8 $\alpha$ and $K_{\alpha }$

We have chosen the half saturation of the resource to be 4 times the mean nectar load of Bumble bees (Allen et al. 1978):

$$\begin{aligned} \alpha = 100~\mathrm {mg}\ \end{aligned}$$

and the carrying capacity of the resource $K_{\alpha }$ to be 10 times the half saturation ($\alpha$):

$$\begin{aligned} K_{\alpha }=1000~\mathrm {mg} \end{aligned}$$

In our model, the half saturation ($\alpha$) is related to the ability of Bumble bees to detect the resource: An increase in resource quantity in a patch where the resource quantity is already much larger than $\alpha$ can not be detected by model bees.

In our work, we have considered that bees can not distinguish the quantity of resource in a specific location when the quantity is 4 times larger than their nectar mean load.

A.9 $k _{HU}$

In Allen et al. (1978) it is said that the mean nectar load per bee is 25mg and the mean time spend per boat is 30 min. With this information, we can compute the rate of nectar intake in the nest:

$$\begin{aligned} k _{HU} = \frac{25}{0.5} = 50~\mathrm {mg}/\# \mathrm {h}. \end{aligned}$$