Generalized Stressors on Hive and Forager Bee Colonies

Abstract

Hive-forming bees play an integral role in promoting agricultural sustainability and ecosystem preservation. The recent worldwide decline of several species of bees, and in particular, the honeybee in the United States, highlights the value in understanding possible causes. Over the past decade, numerous mathematical models and empirical experiments have worked to understand the causes of colony stress, with a particular focus on colony collapse disorder. We integrate and enhance major mathematical models of the past decade to create a single, analytically tractable model using a traditional disease modeling framework that incorporates both lethal and sublethal stressors. On top of this synthesis, a major innovation of our model is the generalization of stressor attributes including their transmissibility, impairment level, lethality, duration, and temporal-occurrence. Our model is validated against numerous emergent, biological characteristics and demonstrates that precocious foraging and labor destabilization can produce colony collapse disorder. The thresholds for these phenomena to occur depend on the characteristics and timing of the stressor, thus motivating further empirical and theoretical studies into stressor characteristics.

Appendices

Mathematical Derivations

As a reminder, we will always assume that stressor-independent parameters and $\nu$ are strictly positive, and the remaining stressor-dependent parameters ($\beta$, $\rho$, and p) are non-negative. Further, we remind the reader that $\mathbf {\theta }$ is defined as a grouping of all model parameters. The GitHub repository contains a Mathematica notebook with the direct computations of many of the reference quantities in these results.

Lemma 1

$E_2$ and $E_3$ are biologically feasible (real and non-negative in each component) if and only if $\psi > 0$ and $w>0$ is sufficiently small.

Proof

Letting w be sufficiently small guarantees that each component of $E_2$ and $E_3$ are real. Further we note that $\text {sgn}(H) > 0$ will guarantee that $E_2$ and $E_3$ are non-negative in each component. Since

$$1 + \sqrt{1 - \frac{4w \gamma \delta \sigma (\beta \nu + \mu \phi )\phi ^2}{K \psi ^2}} \ge 1 - \sqrt{1 - \frac{4w \gamma \delta \sigma (\beta \nu + \mu \phi )\phi ^2}{K \psi ^2}} > 0,$$

we conclude that $\psi > 0$ is a necessary and sufficient condition for the biological feasibility of $E_2$ and $E_3$. $\square$

Theorem 1

Solutions to the system of equations 6 with initial conditions, $(H(t), F_U(t), F_I(t))\big |_{t=0} \in \mathcal {D} = \{\mathbb {R}^+ \times \mathbb {R}^+ \times \mathbb {R}^+\}$ are contained in $\mathcal {D}$.

Proof

Define $dH/dt = f_1\left( H,F_U, F_I\right)$, $dF_U/dt = f_2\left( H,F_U, F_I\right)$, and $dF_I/dt = f_3\left( H,F_U, F_I\right)$ where $f_1,f_2,$ and $f_3$ are the right hand sides of the differential equations listed in system 6. Then by inspection, $f_1, f_2, f_3$ are all continuous in $\mathcal {D}$. Furthermore, by inspecting the Jacobian of the system defined by 6, we can conclude that $\partial f_i/\partial H$, $\partial f_i/\partial F_U$, $\partial f_i/\partial F_I$, for $i \in \{1,2,3\}$ are continuous in $\mathcal {D}$. Since $\mathcal {D}$ is an open, connected, subset of $\mathbb {R}^n$, then by the Existence and Uniqueness Theorem in Strogatz (2018), we conclude that solutions to the system defined by 6 exist and are unique.

We must now show that solutions are contained in $\mathcal {D}$. Consider the boundary, $\partial \mathcal {D} = \{ 0 \times \mathbb {R}^+ \times \mathbb {R}^+\} \cup \{ \mathbb {R}^+\times 0 \times \mathbb {R}^+\} \cup \{\mathbb {R}^+ \times \mathbb {R}^+ \times 0\} = \partial \mathcal {D}_1 \cup \partial \mathcal {D}_2 \cup \partial \mathcal {D}_3$. Since $f_1|_{\partial \mathcal {D}_1}, f_2|_{\partial \mathcal {D}_2}, f_3|_{\partial \mathcal {D}_3} > 0$, then by the differentiability and thus continuity of the solutions to the system defined by 6 we conclude that solutions exist and are contained within $\mathcal {D}$. $\square$

Theorem 2

With complete impairment ($p=0$), the extinction equilibrium, $E_1$, is locally asymptotically stable.

Proof

Let ${\varvec{J}}(E_1; \mathbf {\theta })$ denote the Jacobian of the system defined by 6 evaluated at the extinction equilibrium, $E_1$. Let $P(\lambda )$ denote the characteristic polynomial of ${\varvec{J}}(E_1; \mathbf {\theta })$. By direct computation of

$$\begin{aligned} P(\lambda ) = \lambda ^3 + a_1\lambda ^2 + a_2\lambda + a_3, \end{aligned}$$

we observe that $a_1, a_3 >0$ and $a_1 a_2 > a_3$. Hence, by the Routh-Hurwitz criterion Kot (2001), $E_1$ is asymptotically stable. $\square$

Theorem 3

With complete impairment ($p=0$) and sufficiently small perturbation term, w, the persistence equilibrium, $E_2$, is unstable.

Proof

Let ${\varvec{J}}(E_2; \mathbf {\theta })$ denote the Jacobian of the system defined by 6 evaluated at the persistence equilibrium, $E_2$. Let $P(\lambda )$ denote the characteristic polynomial of ${\varvec{J}}(E_2; \mathbf {\theta })$. By direct computation of

$$\begin{aligned} P(\lambda ) = \lambda ^3 + a_1\lambda ^2 + a_2\lambda + a_3, \end{aligned}$$

we have that $a_3 < 0$ by inspection. Thus, since $P(0) < 0$, $\lim \limits _{\lambda \rightarrow \infty } P(\lambda ) = \infty$, and since $P(\lambda )$ is a continuous function in $\lambda$, it follows by the Intermediate Value Theorem that there exists a positive root of $P(\lambda )$, meaning a positive eigenvalue, ensuring that $E_2$ is unstable. $\square$

Theorem 4

With complete impairment ($p=0$), sufficiently small perturbation term, w, and $\psi > 0$, the persistence equilibrium, $E_3$, is locally asymptotically stable.

Proof

Let ${\varvec{J}}(E_3; \mathbf {\theta })$ denote the Jacobian of the system defined by 6 evaluated at the persistence equilibrium, $E_3$. Let $P(\lambda )$ denote the characteristic polynomial of ${\varvec{J}}(E_3; \mathbf {\theta })$. By direct computation of

$$\begin{aligned} P(\lambda ) = \lambda ^3 + a_1\lambda ^2 + a_2\lambda + a_3, \end{aligned}$$

we have that $a_1, a_3 > 0$ by inspection. Define $Q = a_1a_2 - a_3$. Q is continuously differentiable in $\gamma$, and $dQ/d\gamma > 0$. We now compute the value of $\gamma$ such that $\psi = 0$, namely, $\gamma ^*$. Since $\psi$ is linear and increasing in $\gamma$, then $\psi> 0 \implies \gamma > \gamma ^*$. Since $Q|_{\gamma = \gamma ^*} > 0$ we conclude that $Q > 0$ for all $\psi > 0$. Thus, $a_1 a_2 > a_3$. Hence, by the Routh-Hurwitz criterion Kot (2001), $E_3$ is asymptotically stable. $\square$

Corollary 1

$E_3$ is biologically feasible and asymptotically stable if and only if $\gamma > \mu \left( a + \frac{\sigma }{\mu + c\sigma }\right)$ and $\beta < \frac{1 + \rho /\nu }{2a}\left( \gamma - 2a \mu - (a c + 1)\sigma + \sqrt{\gamma ^2 + 2(ac-1)\gamma \sigma + \sigma ^2(ac + 1)^2}\right)$

Proof

$E_3$ is biologically feasible if and only if $\psi > 0$ by Lemma 1, and thus it’s feasibility guarantees its asymptotic stability by Theorem 4.

We next seek equivalent conditions to $\psi > 0$. $\psi$ can be expressed as,

$$\begin{aligned} \psi (\beta ) = a_0\beta ^2 + a_1 \beta + a_2. \end{aligned}$$

The signs of $a_0$, $a_1$, and $a_2$ can be determined based on the value of $\gamma$.

Now we show that $\psi > 0$ is only possible for sufficiently large $\gamma$. By Descartes’ rule of signs (see Table 2) if $\gamma < \mu \left( a + \frac{\sigma }{\mu + c\sigma }\right)$ then $\psi (0) < 0$ and there can be no positive real roots. Hence, we require $\gamma > \mu \left( a + \frac{\sigma }{\mu + c\sigma }\right)$, in which case we are guaranteed that $\psi (\beta )$ has exactly one positive real root. Since $\psi (0) > 0$ in this case, by continuity it is also true that $\psi (\beta ) > 0$ for sufficiently small $\beta$. We can now use the quadratic formula to solve for the roots of $\psi$ in $\beta$, taking the larger of the two as the upper-bound on $\beta$ for positivity of $\psi$,

$$\begin{aligned} \beta < \frac{1 + \rho /\nu }{2a}\left( \gamma - 2a \mu - (a c + 1)\sigma + \sqrt{\gamma ^2 + 2(ac-1)\gamma \sigma + \sigma ^2(ac + 1)^2}\right) . \end{aligned}$$

$\square$

Table 2 Sign of each coefficient of $\psi (\beta )$ for various values of $\gamma$

Parameter Estimation

We assume $\gamma$ to be the maximum per-capita growth rate achieved in a resource abundant and stress-free environment so that the eclosion rate becomes $E \approx \gamma$. A 22-day study of brood production before adult emergence (Harbo 1986) was used to minimize additional effects (e.g. social inhibition, forager dependence) to estimate $\gamma$ under these conditions. Our model further accounts for brood fatality (see estimation for a) therefore we ignore those effects in the estimation of $\gamma$. Further, since we do not explicitly model the brood stages we elect to use a brood maturation time of 22 days, as used by Harbo (1986) to measure population count. Converting 22-day brood production into a daily rate (see Table 3) and averaging the different colonies, we find $\gamma \approx 0.0625$ with a standard deviation of 0.0228. We use two standard deviations around the point estimate for our LHS range.

Table 3 Data used to estimate $\gamma$ from Harbo (1986)

Next we estimate the proportion of hive bees to forager bees that produces a net-zero flux through social inhibition. Studies conducted by Klein et al. (2019) found that on average, at least 27% of bees foraged in some essence. Assuming this population is under the condition that $H=cF$, we then calculate that

$$\begin{aligned} 0.27 = \frac{F}{N} = \frac{F}{H + F} = \frac{F}{cF + F} = \frac{1}{c + 1} \implies c = 2.70. \end{aligned}$$

Next we attempt to calculate the saturation constant in the dependency of reproduction of hive bees to the presence of forager bees, a. Similar to our estimation of $\gamma$, we assume that the rate of eclosion approaches its maximum rate when the population has few pressures so that

$$\begin{aligned} E \approx b\gamma \end{aligned}$$

where $b = 1 - \epsilon$ for $\epsilon > 0$ where $\epsilon$ is small, quantifies the inability of bees to precisely meet their maximum reproduction rate (due to brood fatality, etc). Expanding the above approximation,

$$\begin{aligned} \gamma \frac{F}{acF + F} \approx b\gamma \implies \frac{1}{ac + 1} \approx b \implies \frac{1-b}{cb} \approx a. \end{aligned}$$

(21)

Egg and larval mortality can be from 5 to 20% (Harbo 1986), therefore we elect to vary b and recalculate our estimate for a when conducting sensitivity analysis.

Studies by Rueppell et al. (2009) measured the flightspan (time from first flight to last appearance) of forager bees across different colony sizes. The average lifespan recorded by Rueppell et al. (2009) across all colony sizes was approximately 7.375 days (see Table 4), which is equivalent to $1/\mu$ in the model. Using the widest possible 95% confidence interval from Rueppell et al. (2009), we generate an LHS range of $6.0 \le 1/\mu \le 9.4$.

Table 4 Data used to estimate $\mu$ from Rueppell et al. (2009)

Fig. 6

Relationship between each parameter and the metrics: colony size at 1-year (solid line) and colony size at 5-years (dashed line). Each parameter is allowed to vary at 15 distinct values while all other parameters remain at default value. Initial conditions used where $(H,F_U,F_I)|_{t=0} = (200,50,0)$

Our model assumes that in the absence of foragers, hive bees will mature to forager bees at rate of $S \approx \sigma H$, meaning an average duration spent in the hive class of $1/\sigma$. The minimum age at which hive bees can become forager bees has been estimated as 2 (Perry et al. 2015), therefore we assume $\sigma \le 0.5$. Further, AAOF in colonies were social inhibition is not negligible is on average, no less than 18.6 days (Rueppell et al. 2009), therefore $1/\sigma \le 1/S \approx 18.6$ yields $\sigma \ge 0.0538$ with plots provided in Appendix B (Figs. 6, 7).

Fig. 7

Relationship between each parameter and the metrics: AAOF at 1-year (solid line) and AAOF at 5-years (dashed line). Each parameter is allowed to vary at 15 distinct values while all other parameters remain at default value. Simulations with terminal populations of less than 100 bees are omitted since AAOF is not defined. Initial conditions used where $(H,F_U,F_I)|_{t=0} = (200,50,0)$

Social Inhibition

The functional form of social inhibition used by Khoury et al. (2011), adjusted for our notation, was,

$$\begin{aligned} S= \left( \alpha - \sigma \left( \frac{F}{H+F_U}\right) \right) H. \end{aligned}$$

This functional form is undefined for $H = F_U = 0$, and further, does not exhibit monotonic behavior as class balance increases, see Fig. 8.

Fig. 8

Comparing social inhibition functional forms from Khoury et al. (2011) (left) and this study (right) with complete impairment ($p = 0$). Default parameters from the respective studies were used (reference Khoury et al. (2011) and Table 1). Color indicates relative social inhibition flow rate to maximum over the domain of the (H, F) space plotted against (Color figure online)