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arXiv:1505.03742v1 [math.DS] 14 May 2015
Disease dynamics of Honeybees with Varroa destructor as parasite and
virus vector
Yun Kang1, Krystal Blanco2, Talia Davis 3, and Ying Wang 4
Abstract
The worldwide decline in honeybee colonies during the past 50 years has often been linked to the spread
of the parasitic mite Varroa destructor and its interaction with certain honeybee viruses carried by Varroa
mites. In this article, we propose a honeybee-mite-virus model that incorporates (1) parasitic interactions
between honeybees and the Varroa mites; (2) five virus transmission terms between honeybees and mites at
different stages of Varroa mites: from honeybees to honeybees, from adult honeybees to phoretic mites, from
honeybee brood to reproductive mites, from reproductive mites to honeybee brood, and from honeybees to
phoretic mites; and (3) Allee effects in the honeybee population generated by its internal organization such as
division of labor. We provide completed local and global analysis for the full system and its subsystems. Our
analytical and numerical results allow us have a better understanding of the synergistic effects of parasitism
and virus infections on honeybee population dynamics and its persistence. Interesting findings from our work
include: (a) Due to Allee effects experienced by the honeybee population, initial conditions are essential for
the survival of the colony. (b) Low adult honeybee to brood ratios have destabilizing effects on the system,
generate fluctuated dynamics, and potentially lead to a catastrophic event where both honeybees and mites
suddenly become extinct. This catastrophic event could be potentially linked to Colony Collapse Disorder
(CCD) of honeybee colonies. (c) Virus infections may have stabilizing effects on the system, and could make
disease more persistent in the presence of parasitic mites. Our model illustrates how the synergy between the
parasitic mites and virus infections consequently generates rich dynamics including multiple attractors where
all species can coexist or go extinct depending on initial conditions. Our findings may provide important
insights on honeybee diseases and parasites and how to best control them.
Keywords: Allee Effects; Honeybees; Extinction; Virus; Parasite; Colony Collapse Disorder (CCD)
1. Introduction
Honeybees are the world’s most important pollinators of food crops. It is estimated that one third of food
that we consume each day mainly relies on pollination by bees. For example, in the United States, honeybees
are major pollinators of alfalfa, apples, broccoli, carrots and many other crops, and hence are of economic
importances. Honeybees have an estimated monetary value between $15 and $20 billion dollars annually as
commercial pollinators in the U.S [23]. There are growing concerns both locally and globally that despite
a 50% growth in honeybee stocks, the supply cannot keep up with the over 300% increase in agricultural
demands [66]. Therefore, the recent sharp declines in honeybee populations have been considered as a global
crisis. The most recent data from the 2012-2013 winter has shown an average loss of 44.8% of hives in the
U.S., and a total of 30.6% loss of commercial hives [55]. Some beekeepers have reported a lost of as many
1Sciences and Mathematics Faculty, College of Letters and Sciences, Arizona State University, Mesa, AZ 85212, USA
(yun.kang@asu.edu)
2Simon A. Levin Mathematical and Computational Modeling Sciences Center, Arizona State University, Tempe, AZ 85281,
USA
3School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85281, USA
4School of Life Sciences, Arizona State University, Tempe, AZ 85281, USA.
Preprint submitted to XXX May 16, 2015
as 90% of their hives [18,39].
Between 1972 and 2006, the wild honeybee populations declined severely and are now considered vir-
tually nonexistent [38,63]. Hence the use of commercial honeybees for pollination is extremely important.
Beginning in 2006, beekeepers began to report an unusual phenomenon in dying bee colonies. Worker bees
would leave the colony to forage and never return, leaving the queen and the young behind to die. No
dead worker bees were found at the nest sites; they simply disappear [11,58]. This phenomenon is known
as Colony Collapse Disorder (CCD), which is a serious problem threatening the health of honey bees and
therefore the economic stability of commercial beekeeping and pollination operations.
The exact causes and triggering factors for CCD have not been completely understood yet. Researchers
have proposed several possible causes of CCD including stress on nutritional diet, harsh winter conditions,
lack of genetic diversity, exposure to certain pesticides, diseases, and parasitic mites Varroa destructor which
are also vectors of viral diseases of honeybees [23,44]. Even before CCD was detected in honeybee colonies,
studies showed that most of the loses could be generally attributed to two main causes: the vampire mite,
Varroa destructor, which feeds on host haemolymph, weakens host immunity and exposes the bees to a
variety of viruses, and the tracheal mite, which infests the breathing tubes of the bee, punctures the tracheal
wall and sucks the bee’s blood and also exposes the bee to a variety of viruses [47,53,34]. Since then,
Varroa mites have been implicated as the main culprit in dying colonies. For example, in Canada, Varroa
mites have been found to be the main reason behind wintering losses of bee colonies [21], and more generally
studies have shown that if the mite population is not properly controlled, the honeybee colony will die [52].
Recent studies also suggest that the Varroa mite could be a contributing cause of CCD since they not only
ectoparasitically feed on bees, but also vertically transmit a number of deadly viruses to the bees [29,28].
There have been at least 14 viruses found in honeybee colonies [2,29], which can differ in intensity of impact,
virulence, etc. for their host. For example, the Acute Bee Paralysis Virus (ABPV) affects the larvae and
pupae which fail to metamorphose to adult stage, while in contrast the Deformed Wing Virus (DWV) affects
larvae and pupae, which can still survive to the adult stage [59].
Mathematical models are powerful tools that could help us obtain insights on potential ecological processes
that link to CCD, and important factors that contribute to the mortality of honeybees. Few sophisticated
mathematical models of honeybee populations have been previously developed. DeGrandi-Hoffman et al.
[14] produced the first time-based honeybee colony growth model. Martin [33] developed a simulation model
consisting of ten components, which linked together various aspects of mite biology using computer software
(ModelMaker); and Martin [34] later extended this model by including a bee model adapted from [14] to
explain the link between the Varroa mite and collapse of the host bee colony. Wilkinson and Smith [65]
proposed a difference equation model of Varroa mites reproducing in a honeybee colony. Their study focused
on parameter estimations and sensitivity analysis. Simulation models are useful but may be too complex to
study mathematically and obtain general predictions.
More recently, mathematical models have been formulated to explore potential mechanisms causing CCD
to the honeybee colony. Sumpter and Martin [56] modeled the effects of a constant population of Varroa
mites on the brood and on adult worker bees, and found that sufficiently large mite infestations may make
hives vulnerable to collapse from viral epidemics. Eberl et al. [16] developed a model connecting Varroa
mites to CCD by including brood maintenance terms which reflect that a certain number of worker bees
is always required to care for the brood in order for them to survive. They found an important threshold
for the number of hive worker bees needed to maintain and take care of the brood. Khoury et al. [26,25]
developed differential equations models to study different death rates of foragers and the impact it had on
colony growth and development. They then linked their results to CCD. Betti et al. [5] studied a model
that combines the dynamics of the spread of disease within a bee colony with the underlying demographic
dynamics of the colony to determine the ultimate fate of the colony under different scenarios. Their results
suggest that the age of recruitment of hive bees to foraging duties is a good early marker for the survival or
2
collapse of a honeybee colony in the face of infection. Kribs-Zaleta et al. [27] created a model to account for
both healthy hive dynamics and hive extinction due to CCD, modeling CCD via a transmissible infection
brought to the hive by foragers. Perry et al. [43] examined the social dynamics underlying the dramatic
colony failure with an aid of a honeybee population model. Their model includes bee foraging performance
varying with age, and displays dynamics of colony population collapse that are similar to field reports of
CCD. These models, no doubt, are insightful and provide us a better understanding on the potential mecha-
nisms that link to CCD. However, most of these models only account for the honeybee population dynamics
with mites or viruses but not both.
The host-parasite relationship between honeybees and Varroa mites has been complicated by the mite’s
close association with a wide range of honeybee viral pathogens. In order to understand how Varroa mite in-
festations and the related viruses transmitted to honeybees affect honeybee population dynamics, and which
may link to CCD, there is a need to develop realistic and mathematically tractable models that include
both mite and pathogen population dynamics. The goal of our work is to develop a useful honeybee-mite-
virus system to obtain better understanding on the synergistic effects of honeybee-mite interactions and
honeybee-virus interactions on the honeybee populations dynamics, thus develop good practices to control
these parasites to maintain or increase honeybee population. The most relevant modeling papers for our
study purposes are by Sumpter and Martin [56], and Ratti et al. [44] whose work examined the transmission
of viruses via Varroa mites, using the susceptible-infectious (SI) disease modeling framework with mites as
vectors for transmission. However, Sumpter and Martin assumed that the mites’ population is constant
while Ratti et al. took no account of the fact that virus transmissions occur at different biological stages of
Varroa mites and honeybees.
In this article, we follow both approaches of Sumpter and Martin [56] and Ratti et al. [44], and propose a
honeybee-mite-virus model that incorporates (1) parasitic interactions between honeybee and Varroa mites;
(2) different virus transmission terms that account for the virus transmission among honeybees, between
honeybees and mites at different stages of Varroa mites; and (3) Allee effects in the honeybee population
generated by the internal organization of honeybees, including division of labor. Our proposed model will
allow us explore the following questions:
1. What are the dynamics of a system only consisting of honeybees and the disease?
2. What are the dynamics of a system only consisting of honeybees and Varroa mites?
3. What are the synergistic effects of Varroa mites and the disease on the honeybee population, and how
may these synergistic effects contribute to CCD?
4. How can we maintain honeybee populations?
The structure of the remaining article is organized as follows: In Section 2, we first provide the biological
background of honeybees, Varroa mites, and the associated virus transmission routes in the honeybee-mite
system; then we derive our SI-type model for honeybees co-infected with the mite and virus. In Section
3, we perform local and global analysis of the proposed model and the related subsystems. The results
from the analysis are then connected to biological contexts and implications. Additionally, we also explore
numerical simulations of the subsystems and the full system to obtain the effects of each parameter in our
system. In Section 4, we summarize our results and the related biological implications of our studies in
finding potential causes of Colony Collapse Disorder. We also provide potential projects for future work.
The detailed mathematical proofs of our theoretical results are provided in the last section.
3
2. Biological background and model derivations
Honeybee colony: During the spring and summer, a honeybee colony typically consists of a single
reproductive queen, 20,000 – 60,000 adult worker bees, 10,000 – 30,000 individuals at the brood stage (egg,
larvae and pupae) and up to hundreds of male drones. During the winter, the colony typically reduces in size
and consists of a single queen and somewhere between 8,000 – 15,000 worker bees [34]. A large population
of workers carry out the tasks of the bee colony, which include foraging, pollination, honey production and,
in particular, caring for the brood and rearing the next generation of bees. The queen is the only fertile
individual of the colony and has an average life span of 2 – 3 years [56]. During the peak season (in the
summer), the queen lays up to 2000 eggs per day, where fertilized eggs produce female worker bees, or much
more rarely queens, while drones develop from non-fertilized eggs [5]. The bees go through the following
stages in development: egg (about 3 days), larvae (about 7 days), pupae (about 14 days), and adult. The
life span of an adult worker bee also depends on the season. Workers usually have a lifespan of 3 – 6 weeks
during the spring and summer, and are reported to live as long as 4 months during the winter [40]. The
adult drone life span is typically 20 – 40 days, with reports of drone living up to 59 days under optimal
colony conditions [40,22].
Let Nh(t) be the total number of honeybees in the colony, including the larvae, pupae and adult bee
(both hives and foragers) at time t. Let us assume that the honeybee colony has (1 −ξh)Nhpopulation of
brood, i.e., the ratio of adult honeybees to the brood is captured by ξh
1−ξhwhere ξh∈[0,1]. In general, the
ratio ξhvaries with time, and the successful honeybee colony should have ξh
1−ξh>2 [49]. For convenience,
we assume that ξhis a fixed constant. This implies that the honeybee colony has ξhNhadult honeybees and
(1 −ξh)Nhbrood. In the absence of mites and virus, the population dynamics of Nh(t) is described by the
following nonlinear equation:
N′
h=r(ξhNh)2
K+ (ξhNh)2−dhNh(1)
where ris the maximum birth rate, specified as the number of worker bees born per day; the parameter
√Kis the size of the bee colony at which the birth rate is half of the maximum possible rate; and dhis
the average death rate of the worker honeybees. The term (ξhNh)2
K+(ξhNh)2describes that the successful survival
of an egg which will develop into a worker bee needs the care of adult honeybees (ξhNh) inside the colony
and also needs food brought in by the honeybee foragers. This approach follows the modeling idea in [16]
for honeybee diseases and in [24] for the population of leaf-cutter ants. This term implicitly includes the
internal organization of the honeybee population, such as division of labor.
Varroa mites: Varroa mites were first reported in Kentucky in 1991. They have since spread to become
a major pest of honeybees in many states [4]. Varroa mites are external honeybee parasites that attack
both adult honeybees and brood, with a distinct preference for drone brood [41]. They suck the blood from
both the adults and the developing brood, weakening them and shortening the life span of the bees which
they feed on. Emerging brood may be deformed, and may be born with missing legs or wings. Untreated
infestations of Varroa mites can cause honeybee colonies to collapse [32].
The mites go through a series of stages: larva, protonymph, deutonymph and then adult. Adult females
undergo two phases in their life cycle, the phoretic and reproductive phases. During the phoretic phase,
female Varroa feed on adult bees and are passed from bee to bee as they pass one another in the colony.
During the phoretic phase, the female Varroa mites live on adult bees and can usually be found between the
abdominal segments of the bees. The mites puncture the soft tissue between the segments and feed on bee
hemolymph, harming the host [46,6]. Mite reproduction can occur only if honeybee brood is available. A
female mite enters the brood cell about one day before capping and will be sealed in with the larva. After
the capping of the cell, it lays a single male egg and several female eggs at 30-hour intervals [60], and the
4
mite feeds and develops on the maturing bee larva. When the host bee leaves the cell, the mature female
mites leave the cell. The male mite dies after mating with his sisters, and if immature female mites are
present they die as they come out of the cell, as they cannot survive once outside the cell. The adult female
mite begins searching for other bees or larvae to parasitize.
The phoretic period of the mite appears to contribute to the mite’s reproductive ability, which may last
4.5 to 11 days when brood is present in the hive or as long as five to six months during the winter when little
no brood is present in the hive. Consequently, female mites living when brood is present in the colony have
an average life expectancy of 27 days, yet in the absence of brood, they may live for many months. In the
average temperate climate, mite populations can increase 12-fold in colonies which have brood half of the
year and 800-fold in colonies which have brood year-round. This makes the mites very difficult to control,
especially in warmer climates where colonies maintain brood year-round [19].
Let Nm(t) be the number of adult female Varroa mites in the honeybee colony in the absence of a virus.
Varroa mites feed on the haemolymph of brood and adult honeybees, and their reproduction depends on the
availability of the brood and the population of the reproductive female Varroa mites. Similarly, we define a
fixed ξm∈[0,1] ratio of phoretic stage of Varroa mites to their whole population. We model the parasitic
interactions between Varroa mites and honeybees using the Holling Type I functional responses, i.e.:
ˆα
|{z}
The parasitism rate
(1 −ξh)Nh
|{z }
The honeybee brood p opulation
(1 −ξm)Nm
| {z }
The reproductive female Varroa mites population
=αNhNm
where α= ˆα(1 −ξh)(1 −ξm). This implies that the virus-free system of Varroa mites and the honeybee
population can be described by the following two nonlinear equations:
N′
h=r(ξhNh)2
K+(ξhNh)2−dhNh−αNhNm
N′
m=cαNhNm−dmNm
(2)
where the parameter αmeasures the parasitic rate of Varroa mites; cis the conversion rate from honeybee
consumption to sustenance for reproduction; and dmis the death rate of Varroa mites. Model (2) implies
that Varroa mites population Nmgoes extinct if the population of honeybees Nhgoes extinct.
Varroa mites as a disease-vector for virus transmissions: Varroa mites not only feed on host
haemolymph and weaken host immunity, but they also expose honeybee colonies to at least 14 different
viruses including deadly viruses such as ABPV and DWV. Varroa mites can transmit viruses in their repro-
duction phase to honeybee brood and during the phoretic phase to adult honeybees. To model the virus
transmission between Varroa mites and honeybees during these two phases, we let Sh(t), Sm(t) be the suscep-
tible population of honeybees and Varroa mites, respectively; and Ih(t), Im(t) be the infected population of
honeybees and Varroa mites, respectively. Then the total population of honeybees is Nh(t) = Sh(t) + Ih(t),
and the total population of Varroa mites is Nm(t) = Sm(t) + Im(t).
The virus transmission between female Varroa mites and honeybees can occur in the following two phases
of the Varroa mite life cycle:
1. The honeybee colony has ξhShsusceptible adult honeybees; ξhIhinfected adult honeybees, ξmSm
susceptible phoretic female Varroa mites; and ξmIminfected phoretic female Varroa mites. In the
phoretic phase, female Varroa mites move between adult bees both spontaneously and just prior to
the death of their host bee [56]. Following the approach of [34], we assume that virus transmission is
frequency dependent, i.e.,
•The rate at which susceptible adult honeybees are infected by the infected phoretic female Varroa
5
mites (IPFM) is:
βmh
|{z}
probability being infected after contacts
×ξhSh
|{z}
population of healthy adult honeybees
×ξmIm
ξmSm+ξmIm
|{z }
probability being contacted by IPFM
=βmhξhShIm
Sm+Im.
•The rate at which susceptible phoretic female Varroa mites (SPFM) are infected by the infected
adult honeybees (IAH) is:
βhm
|{z}
probability being infected after contacts
×ξmSm
|{z}
population of SPFM
×ξhIh
ξhSh+ξhIh
|{z }
probability being contacted by IAH
=βhmξmSmIh
Sh+Ih.
2. The honeybee colony has (1 −ξh)Shsusceptible honeybee brood; (1 −ξh)Ihinfected honeybee brood,
(1 −ξm)Smsusceptible reproductive female Varroa mites; and (1 −ξm)Iminfected reproductive female
Varroa mites. Chen et al. found that there is a direct relationship between virus frequency and the
number of mites to which honeybee brood were exposed, i.e., the more donor mites that were introduced
per cell, the greater the incidence of virus that was detected in the honeybee brood [8,9]. This implies
that the virus transmission rate between Varroa mites and the honeybee brood during the reproductive
phase of mites is density dependent, i.e., similar to the term that describes the parasitic interaction
between mites and honeybee. Therefore, we have as follows:
•A newborn honeybee becomes infected if it is parasitize by the infected reproductive female Varroa
mites. Thus, the rate at which susceptible honeybee brood is infected by the infected reproductive
female Varroa mites (IRFM) is:
βmh2
|{z}
probability being infected after contacts
×(1 −ξh)Sh
| {z }
population of healthy honeybee brood
׈α(1 −ξm)Im
| {z }
parasitism by IRFM
=βmh2αShIm.
•The reproduction of Varroa mites depends on honeybee brood. The newborn Varroa mites become
infected if either the brood is infected or the female Varroa mites is infected. Thus, based on the
formulation of the host-parasite interaction model (2), the rate at which infected newborn female
Varroa mites (INFM) become infected depends on the parasitic interaction between mites and
honeybees which can be described as follows:
cα [Ih(Sm+Im) + ShIm]
|{z }
the infected newborn Varroa mites with virus
.
The virus transmission among honeybees: The proportion of honeybees which can infect themselves
is also dependent on the total number of susceptible and infected bees present in the colony, and hence
frequency-dependent transmission is used [34], which is described as follows:
6
βh
|{z}
probability being infected after contacts
×Sh
|{z}
the healthy honeybee p opulation
×Ih
Sh+Ih
|{z }
probability of contacting or being contacted by infected honeybees
=βhShIh
Sh+Ih.
The reduced fitness of honeybees due to virus infections: The parasitic Varroa mites have been
shown to act as a vector for a number of viruses including DWV, ABPV, Chronic Bee Paralysis Virus
(CBPV), Slow Bee Paralysis Virus (SPV), Black Queen Cell Virus (BQCV), Kashmir Bee Virus (KBV),
Cloudy Wing Virus (CWV), and Sacbrood Virus (SBV) [33,34,35,28]. These virus infections contribute
to morphological deformities of honeybees such as small body size, shortened abdomen and deformed wings,
which reduce vigor and longevity, and they can also influence flight duration and the homing ability of
foragers [28]. In our model, we assume that the infected adult honeybee population ξhIhcontributes to the
reproduction of honeybees with a reduced rate ρ∈(0,1), therefore, the healthy honeybee population Shcan
be modeled as follows,
S′
h=rξ2
h(Sh+ρIh)2
K+ξ2
h(Sh+ρIh)2
|{z }
reproduction of honeybee s
−βhShIh
Sh+Ih
|{z }
honey bee infected by themselves
−αSh(Sm+Im)
|{z }
parasitism by mites
−βmh(ξhSh)Im
Sm+Im
|{z }
adult honeybees infected by the phoretic mite s
−βmh2αShIm
|{z }
honeybee broo d infected by the reproductive mites
−dhSh
.(3)
And the infected honeybee population can be modeled by the following equation,
I′
h=ShhβhIh
Sh+Ih+βmhξhIm
Sm+Im+βmh2αImi−αIh(Sm+Im)
|{z }
Consumed by mites
−(dh
|{z}
natural honeybee mortality rate
+µh
|{z}
additional death due to virus infections
)Ih
.(4)
Let µmbe the additional death rate of Varroa mites due to virus infections. Then the population of
healthy Varroa mites Smand the infected Varroa mites Imcan be described by the following set of nonlinear
equations:
S′
m=Sm
cαSh−βhmξmIh
Sh+Ih
|{z }
the phoretic mites infected by adult honeybees
−dm
|{z}
natural mortality rate of mi tes
I′
m=cα [Ih(Sm+Im) + ShIm]
|{z }
mites born with virus infectio ns
+βhmIh(ξmSm)
Sh+Ih−(dm+µm
|{z}
additional death due to virus infectio ns
)Im
.(5)
Based on the discussions above, the full model of honeybee-mites-virus population dynamics is therefore
7
modeled by the following system of differential equations:
S′
h=rξ2
h(Sh+ρIh)2
K+ξ2
h(Sh+ρIh)2−dhSh−βhShIh
Sh+Ih−βmh(ξhSh)Im
Sm+Im
−βmh2αIm−αSh(Sm+Im)
I′
h=ShhβhIh
Sh+Ih+βmhξhIm
Sm+Im+βmh2αImi−αIh(Sm+Im)−(dh+µh)Ih
S′
m=SmhcαSh−βhmξmIh
Sh+Ih−dmi
I′
m=cα [Ih(Sm+Im) + ShIm] + βhmIh(ξmSm)
Sh+Ih−(dm+µm)Im
.(6)
For convenience, let ˆ
K=K
ξ2
h
,ˆ
βmh =βmhξh,˜
βmh =βmh2α, ˆ
βhm =βhmξm.Then the full model (6) can
be rewritten as the following model
S′
h=r(Sh+ρIh)2
ˆ
K+(Sh+ρIh)2−dhSh−βhShIh
Sh+Ih−ˆ
βmhShIm
Sm+Im−˜
βmhShIm−αSh(Sm+Im)
I′
h=ShhβhIh
Sh+Ih+ˆ
βmhIm
Sm+Im+˜
βmhImi−αIh(Sm+Im)−(dh+µh)Ih
S′
m=SmhcαSh−ˆ
βhmIh
Sh+Ih−dmi
I′
m=cα [Ih(Sm+Im) + ShIm] + ˆ
βhmIhSm
Sh+Ih−(dm+µm)Im
(7)
where α, ρ, c ∈[0,1] and the virus transmission rates βh,ˆ
βmh,ˆ
βhm,˜
βmh ∈(0,1). In summary, the full
honeybee-mite-virus model (7) incorporates (1) Allee effects of honeybees due to the cooperation of the in-
ternal organization; (2) parasitism interactions between honeybee and mites; (3) the vertical disease transmis-
sion mode modeled by the frequency-dependent disease transmission function during Varroa mites’ phoretic
phase; (4) the horizontal disease transmission mode modeled by the density-dependent disease transmission
function during Varroa mites’ reproductive phase; and (5) the reduced fitness of honeybees due to virus
infections. The rest of this manuscript will focus on the dynamics of Model (7).
3. Mathematical analysis
Let ShIh
Sh+IhSh=Ih=0 = 0 and Im
Sm+ImSm=Im=0 = 0. Define X={(Sh, Ih, Sm, Im)∈R4
+:Sh+Ih>
0 and Sm+Im>0}, then Xcan be considered as the state space of our model (7). To continue the analysis,
let us define Nh=Sh+Ih, Nm=Sm+Imas the total population of honeybees and mites, respectively; and
the following notations:
Nc=
r
d−q(r
d)2−4ˆ
K
2, N∗=
r
d+q(r
d)2−4ˆ
K
2
Sc
h=
r
dh−rr
dh2
−4ˆ
K
2, S∗
h=
r
dh+rr
dh2
−4ˆ
K
2
Nc
h=
r
dh+µh+αN∗−rr
dh+µh+αN∗2
−4ˆ
K/ρ2
2, N∗
h=
r
dh+µh+αN∗+rr
dh+µh+αN∗2
−4ˆ
K/ρ2
2
8
ˆ
Sc
h=
r
dh+βh+ˆ
βmh+( ˜
βmh+α)N∗−sr
dh+βh+ˆ
βmh+( ˜
βmh+α)N∗2
−4ˆ
K
2,
ˆ
S∗
h=
r
dh+βh+ˆ
βmh+( ˜
βmh+α)N∗+sr
dh+βh+ˆ
βmh+( ˜
βmh+α)N∗2
−4ˆ
K
2
˜
Sc
h=
r
dh+βh+ˆ
βmh+( ˜
βmh+α)(N∗−N∗
h)−sr
dh+βh+ˆ
βmh+( ˜
βmh+α)(N∗−N∗
h)2
−4ˆ
K
2,
˜
S∗
h=
r
dh+βh+ˆ
βmh+( ˜
βmh+α)(N∗−N∗
h)+sr
dh+βh+ˆ
βmh+( ˜
βmh+α)(N∗−N∗
h)2
−4ˆ
K
2
where d= min{dh, dm}. Define fb(x, y) =
r
x+q(r
x)2−4ˆ
K/y
2and fb(x, y) =
r
x−q(r
x)2−4ˆ
K/y
2, then we have
∂f b(x, y)
∂x <0,∂f b(x, y)
∂y <0,∂fb(x, y)
∂x >0,∂fb(x, y)
∂y >0
which imply the following inequalities
Sc
h<˜
Sc
h<ˆ
Sc
h<ˆ
S∗
h<˜
S∗
h< S∗
h, Nc≤Sc
h< Nc
h,and N∗
h< S∗
h≤N∗.
Theorem 3.1 (Basic dynamical properties).Assume that all parameters are strictly positive and ρ, c ∈
[0,1]. The model (7)is positively invariant and bounded in the state space X, which is attracted to the
following compact set
C={(Sh, Ih, Sm, Im)∈R4
+: 0 ≤(Sh+Ih) + (Sm+Im) = Nh+Nm≤N∗}
provided that r
d>2pˆ
Kand time is large enough. Moreover, the following statements hold for Model (7):
•If r
2√ˆ
K> dh, then the total population of honeybees Nhis bounded by S∗
h, i.e.,
lim sup
t→∞
Nh(t)≤S∗
h.
If r
2√ˆ
K>dh+µh+αN∗
ρand Nh(0) > Nc
hhold, then the total population of honeybees Nhis persistent,
i.e.,
N∗
h≤lim inf
t→∞ Nh(t)≤lim sup
t→∞
Nh(t)≤S∗
h≤lim sup
t→∞
N(t) = N∗.
•If one of the following inequalities hold
1. r
2√ˆ
K> dh+βh+ˆ
βmh + ( ˜
βmh +α)N∗and Sh(0) >ˆ
Sc
h,or
2. r
2√ˆ
K>max ndh+βh+ˆ
βmh + ( ˜
βmh +α)(N∗−N∗
h),dh+µh+αN∗
ρowith Nh(0) ≥Sh(0) >˜
Sc
h,
then Shis persistent with the following properties:
ˆ
S∗
h≤˜
S∗
h≤lim inf
t→∞ Sh(t)≤lim inf
t→∞ Nh(t)≤lim inf
t→∞ N(t)≤lim sup
t→∞
N(t)≤N∗.
•The extinction equilibrium E0= (0,0,0,0) is always local stable. Moreover, the system (7)converges
to E0globally if dh>r
2√ˆ
Kholds; and it converges to E0locally if the initial population satisfies either
9
N(0) < Ncor Nh(0) < Sc
h.
Notes: The positive invariance and boundedness results from Theorem 3.1 imply that our model is well-
defined biologically. In addition, Theorem 3.1 indicate follows:
1. Initial conditions are important for the persistence of honeybees.
2. The inequality r
2√ˆ
K> dhis a necessary condition for honeybee persistence, i.e., the large intrinsic
growth rate r, small half saturation ˆ
K, and the small death rate of honeybees dh.
3. The small values of disease transmission rates βh,ˆ
βmh,˜
βmh; and small values of mite attacking rate
αare also important for the persistence of the healthy honeybee population Sh.
Recall that d= min{dh, dm}, N c=
r
d−q(r
d)2−4ˆ
K
2, N∗=
r
d+q(r
d)2−4ˆ
K
2and
N∗
h=
r
dh+µh+αN∗+rr
dh+µh+αN∗2−4ˆ
K/ρ2
2.
Theorem 3.1 implies that under proper initial conditions, honeybees can persist if r
2√ˆ
K>dh+µh+αN∗
ρand Nh(0) >
Nc
hholds. Notice that ξhis the ratio of adult honeybees in the colony, and
r
2pˆ
K
=rξh
2√K>dh+µh+αN∗
ρ⇔rξh
2√K−
αqr
d2−4K
ξ2
h
2ρ>dh+µh+αr
2d
ρ
and rξh
2√K−
αr(r
d)2−4K
ξ2
h
2ρis an increasing function of ξh. This implies that the large hives to brood ratio ξh
is important for the persistence of honeybees.
Theorem 3.2 (Persistence and extinction of disease or mites).The following statements hold
•If N∗<dm
αc , then the total population of mite Nmgoes extinct, i.e.,
lim sup
t→∞
Nm(t) = 0
where system (7)is attracted to the mite-free invariant set MF ={(Sh, Ih, Sm, Im)∈R4
+:Sm+Im=
0}, and its dynamics is equivalent to the following two-D model (8)
S′
h=r(Sh+ρIh)2
ˆ
K+(Sh+ρIh)2−dhSh−ShβhIh
Sh+Ih
I′
h=ShβhIh
Sh+Ih−(dh+µh)Ih
.(8)
•If r
2√ˆ
K> dh, S∗
h<dm
αc ,and Sh(0) > Sc
h, then the total population of honeybees persists while the
healthy mite population Smgoes extinct, i.e.,
lim sup
t→∞
Sm(t) = 0
10
where the system (7)is attracted to the healthy-mite-free invariant set H M F ={(Sh, Ih, Sm, Im)∈
R4
+:Sm= 0}and its dynamics is equivalent to the following three-D system (9):
S′
h=r(Sh+ρIh)2
ˆ
K+(Sh+ρIh)2−dhSh−βhShIh
Sh+Ih−ˆ
βmhSh−˜
βmhShIm−αShIm
I′
h=ShhβhIh
Sh+Ih+ˆ
βmh +˜
βmhImi−αIhIm−(dh+µh)Ih
I′
m=cαImhIh+Sh−dm+µm
cα i.
(9)
•Assume that r
2√ˆ
K> dh+βh+ˆ
βmh + ( ˜
βmh +α)N∗and Sh(0) >ˆ
Sc
h.Then the disease I=Ih+Im
persists if the following condition holds:
min nβh,˜
βmh +cα ˆ
S∗
ho
max n(dh+µh),(dm+µm)o≥1.
Notes: The results of the reduced dynamics in Theorem 3.2 can be easily obtained by the theory of asymp-
totically autonomous systems [7]. The detailed proof of our results are provided in the last section.
If d= min{dm, dh}=dm, then we have N∗=
r
dm+q(r
dm)2−4ˆ
K
2<dm
αc ⇔pαc(αcK −r)< dm<r
2√K.
If dm>r
2√K, then the system (7) converges to the extinction equilibrium E0according to Theorem 3.1.
On the other hand, if d= min{dm, dh}=dh, then according to Theorem 3.2, the condition S∗
h=N∗=
r
dh+rr
dh2
−4ˆ
K
2<dm
αc can lead to the extinction of the mite population. Therefore, we can conclude that
large values of the death rate of mites, dm, can lead to the extinction of the whole colony; and large values
of the death rate of mites, dm, small values of mite attacking rate, α, and its energy conversion rate, c, can
lead to either its own extinction or the extinction of the healthy mite population Sm. Here we would like
to point out that it is possible to have the persistence of infected mites while the healthy mite goes extinct
(see the resulting dynamics (9) when the healthy mite goes extinct).
Let Nh= lim inft→∞ Nh(t) and Nm= lim inf t→∞ Nm(t).
1. Assume that Nh>0 under proper initial conditions (see sufficient conditions provided in Theorem
3.1). If Nh>dm+µm
cα and N(0) > Nhhold, then the total population of mites Nmis persistent, i.e.,
lim inf
t→∞ Nm(t)≥Nh−dm+µm
cα .
2. Assume that r
2√ˆ
K>dh+µh+αN∗
ρ, Nh(0) > Nc
hand Nm>0 (under proper initial conditions). If the
following inequality holds, i.e.,
max nβh+ˆ
βhmN∗
ˆ
S∗
h
+cαS∗
h,ˆ
βmhS∗
h
Nm+˜
βmh +cαS∗
ho
min n(αNm+dh+µh),(dm+µm)o<1,
then the system (7) is attracted to the disease-free invariant set DF ={(Sh, Ih, Sm, Im)∈R4
+:
11
Ih+Im= 0}and its dynamics is equivalent to the following two-D model (10)
S′
h=rS2
h
ˆ
K+S2
h−dhSh−αShSm
S′
m=cαShSm−dmSm
.(10)
A detailed argument supporting the statement above has been provided in the last section, which suggest
that small values of its natural death rate, dm, and the additional death rate due to the virus, µm, can
promote the persistence of the mite population.
The results of Theorem 3.2 also suggest that: 1. the persistence of the virus requires a large value for
the disease transmission rate between adult honeybees, βh, or that the disease transmission rate between
honeybee brood and reproductive mites, ˜
βmh; or small values of total death rates of honeybees, dh+µh, and
mites, dm+µm;2. the extinction of the virus requires small values of all disease transmission rates, i.e.,
small values of βh,ˆ
βmh,˜
βmh,ˆ
βhm; or large values of total death rates of honeybees and mites.
In the following three subsections, we explore the global dynamics of the mite-free subsystem (8), the disease-
free subsystem (10), and the healthy-mite-free subsystem (9).
3.1. Dynamics of the mite-free subsystem
First, let us define the following notations:
a=1
βh
dh+µh−1,˜
d= (a+ 1)dh+µh=dhβh
dh+µh
βh
dh+µh−1+µh,
I1
h=
r
˜
d−r(r
˜
d)2−4ˆ
K
(a+ρ)2
2, I2
h=
r
˜
d+r(r
˜
d)2−4ˆ
K
(a+ρ)2
2, Sk
h=aIk
h, k = 1,2.
According to Theorem 3.2, if N∗<dm
αc , then the dynamics of (7) is equivalent to the following mite-free
dynamics (8)
S′
h=r(Sh+ρIh)2
ˆ
K+(Sh+ρIh)2−dhSh−ShβhIh
Sh+Ih
I′
h=ShβhIh
Sh+Ih−(dh+µh)Ih
whose dynamics can be summarized by the following theorem:
Theorem 3.3 (Dynamics of the mite-free subsystem).The mite-free subsystem (8)can have one, three,
or five equilibria. Define RV
0=βh
dh+µh. The existence and stability conditions for these equilibria are listed
in Table 1. In addition, the global dynamics of the mite-free subsystem (8)can be summarized as follows:
1. The trajectory of (8)converges to extinction (0,0) for all initial conditions in R2
+if one of the following
conditions hold:
•r
2√ˆ
K< dh, or
•RV
0>1and dh<r
2√ˆ
K<˜
d
a+ρ.
2. The trajectory of (8)converges to either (0,0) or (S∗
h,0) for almost all initial conditions in R2
+if the
inequalities RV
0<1and r
2√ˆ
K> dh.
3. The trajectory of (8)converges to either (0,0) or (S2
h, I2
h)for almost all initial conditions in R2
+if the
inequalities RV
0>1,r
2√ˆ
K>max ndh,˜
d
a+ρohold.
12
Equilibria Existence Condition for Existence Stability Condition
(0,0) Always exists Always locally stable
(Sc
h,0) r
2√ˆ
K> dhSaddle if RV
0<1; Source if RV
0>1
(S∗
h,0) r
2√ˆ
K> dhSink if RV
0<1; Saddle if RV
0>1
(S1
h, I1
h)RV
0>1 and r
2√ˆ
K>˜
d
a+ρAlways a saddle.
(S2
h, I2
h)RV
0>1 and r
2√ˆ
K>˜
d
a+ρAlways locally asymptotically stable
Table 1: The existence and stability of equilibrium for the mite-free subsystem (8). We have RV
0=βh
dh+µh,a=1
RV
0−1,˜
d=
(a+ 1)dh+µh=dhRV
0
RV
0−1+µh, Sc
h=
r
dh
−rr
dh2−4ˆ
K
2, S∗
h=
r
dh+rr
dh2−4ˆ
K
2,and I1
h=
r
˜
d
−rr
˜
d2−4ˆ
K
(a+ρ)2
2, I2
h=
r
˜
d+rr
˜
d2−4ˆ
K
(a+ρ)2
2, Sk
h=aIk
h, k = 1,2.
Notes: Theorem 3.3 implies that the mite-free subsystem (8) has relatively simple dynamics, i.e., no limit
cycle. The results show the following interesting findings:
1. Honeybees can persist with proper initial conditions if the virus transmission rate among honeybees
βhis not large, i.e., RV
0<1.
2. Both honeybees and the virus can coexist if βhis in the medium range, i.e. RV
0>1 and ˜
d
a+ρ<r
2√ˆ
K
3. However, the large virus transmission rate among honeybees βhcan drive honeybees to extinction.
This occurs when the inequalities RV
0>1 and dh<r
2√ˆ
K<˜
d
a+ρhold.
3.2. Dynamics of the healthy-mite-free subsystem
To continue the study, let us define the following notations:
ˆa=(µh+dh−βh−ˆ
βmh) + q(βh+ˆ
βmh −µh−dh)2+ 4 ˆ
βmh(dh+µh)
2ˆ
βmh
,ˆ
d= (ˆa+ 1)dh+µh,
and
ˆ
I1
h=
r
ˆ
d−r(r
ˆ
d)2−4ˆ
K
(ˆa+ρ)2
2,ˆ
I2
h=
r
ˆ
d+r(r
ˆ
d)2−4ˆ
K
(ˆa+ρ)2
2,ˆ
Sk
h= ˆaˆ
Ik
h, k = 1,2.
According to Theorem 3.2, if r
2√ˆ
K> dh, S∗
h<dm
αc ,and Sh(0) > Sc
h, then the total population of honeybees
persists while the healthy mite population Smgoes extinct, i.e.,
lim sup
t→∞
Sm(t) = 0
where the system (7) is attracted to the healthy-mite-free invariant set HM F ={(Sh, Ih, Sm, Im)∈R4
+:
Sm= 0}and its dynamics is equivalent to the following three-D system (9):
S′
h=r(Sh+ρIh)2
ˆ
K+(Sh+ρIh)2−dhSh−βhShIh
Sh+Ih−ˆ
βmhSh−˜
βmhShIm−αShIm
I′
h=ShhβhIh
Sh+Ih+ˆ
βmh +˜
βmhImi−αIhIm−(dh+µh)Ih
I′
m=cαImhIh+Sh−dm+µm
cα i
.
13
The condition of S∗
h<dm
αc is not the necessary condition for lim supt→∞ Sm(t) = 0 in the full system (7). As
an illustration, we can see that the healthy mites go extinct for the following set of parameter values when
S∗
h>dm
αc :
r= 3.5, ρ =.8,ˆ
K= 20, dh=.15, βh= 0.05,ˆ
βmh =.13,˜
βmh = 0.05,
and
α=.1, µh=.12, c = 0.3, βhm = 0.025, dm=.145, µm= 0.005.
The dynamics of the healthy-mite-free subsystem (9) can be summarized by the following theorem:
Theorem 3.4 (Dynamics of the healthy-mite-free subsystem).If r
2√ˆ
K> dhand S∗
h<dm+µm
cα , then the
population of infected mites goes extinct in the subsystem (9), i.e.,
lim sup
t→∞
Im(t) = 0
which leads to the following mite-free model (11):
S′
h=r(Sh+ρIh)2
ˆ
K+(Sh+ρIh)2−dhSh−βhShIh
Sh+Ih−ˆ
βmhSh
I′
h=ShhβhIh
Sh+Ih+ˆ
βmhi−(dh+µh)Ih.
(11)
The dynamics of the mite-free model (11)can be summarized as follows:
1. If r
2√ˆ
K<ˆ
d
ˆa+ρ, then the mite-free model (11)has only the extinction equilibrium (0,0) which is globally
stable.
2. If r
2√ˆ
K>ˆ
d
ˆa+ρ, then the mite-free model (11)has the extinction equilibrium (0,0) and two interior
equilibria (ˆaˆ
Ik
h,ˆ
Ik
h), k = 1,2where (ˆ
S1
h,ˆ
I1
h) = (ˆaˆ
I1
h,ˆ
I1
h)is always a saddle; and both (0,0) and (ˆ
S2
h,ˆ
I2
h) =
(ˆaˆ
I2
h,ˆ
I2
h)are always locally asymptotically stable.
Notes: Theorem 3.4 implies that, under the condition of S∗
h<dm+µm
cα , the infected mite population Im
goes extinct in the healthy-mite-free subsystem (9) which reduces to the new mite-free subsystem (11). The
subsystem (11) occurs from the healthy-mite-free subsystem (9) when increases the value of dm+µm
cα , e.g.,
increasing the value of the natural death rate of mites dm, or increasing the additional death rate of mites
caused by virus infections, or decreasing values of the energy conversion rate c, or decreasing the value of
mite attacking rate α.
Synergistic effects of parasitic mites and virus infections: We have two types of honeybee-virus
interaction subsystems (i.e., no mites): if the colony has no mites at all, then we have the reduced subsystem
(8); if the colony has mites initially but healthy mites go extinct first, and then the infected mites go extinct
during the process due to changes in environment, then we have the limiting subsystem consisting of only
honeybees and the virus (11). From Theorem 3.3, we can conclude that the colony can have persistence of
honeybees while the virus goes extinct when the inequalities RV
0>1,r
2√ˆ
K>max ndh,˜
d
a+ρohold. However,
according to Theorem 3.4, the subsystem (11) has persistence of disease whenever r
2√ˆ
K>ˆ
d
ˆa+ρholds. This
implies that the existence of both mites and virus initially could promote the persistence of disease.
The results from Theorem 3.4 provide us a general view of the dynamics of the healthy-mite-free subsystem
(9). In the rest of this subsection, we will explore the existence condition for the interior equilibrium of (9).
14
An interior equilibrium (Sh, Ih, Im) of the healthy-mite-free subsystem (9), satisfies the following equations:
cαImhIh+Sh−dm+µm
cα i= 0 ⇒Nh=Ih+Sh=dm+µm
cα
r(Sh+ρIh)2
ˆ
K+(Sh+ρIh)2−dhNh−αNhIm−µhIh= 0 ⇒r(Sh+ρIh)2
ˆ
K+(Sh+ρIh)2−dhNh−αNhIm−µhIh= 0
⇒r(Nh−Ih+ρIh)2
ˆ
K+(Nh−Ih+ρIh)2−dhNh−αNhIm−µhIh= 0
⇒Im=
r(dm+µm
cα −Ih+ρIh)2
ˆ
K+(dm+µm
cα −Ih+ρIh)2−dhdm+µm
cα −µhIh
α(dm+µm
cα )=f1(Ih)
(12)
ShhβhIh
Sh+Ih+ˆ
βmh +˜
βmhImi=αIhIm+ (dh+µh)Ih
dm+µm
cα −Ihh βhIh
dm+µm+ˆ
βmh +˜
βmhImi=αIhIm+ (dh+µh)Ih
⇒Im=(dm+µm
cα −Ih)hβhIh
dm+µm+ˆ
βmhi−(dh+µh)Ih
αIh−(dm+µm
cα −Ih)˜
βmh =f2(Ih)
.(13)
The equations above imply that the interior equilibrium of (9) is the positive intercept of f1(Ih) and f2(Ih)
subject to 0 < Ih<dm+µm
cα . The expression for the function f1(Ih) implies that the subsystem (9) has no
interior equilibrium if
r < dh(dm+µm)
cα ⇔rcα < dh(dm+µm)
since Im=f1(Ih)<0 when this inequality holds. Therefore, we could expect the extinction of Imfor small
values of r, c, α and large values of dh, dm, µm. This has been confirmed by numerical simulations. Our
simulations provide important insights on the complicated dynamic of (9), which suggest the following:
•The population of honeybees and infected mites in (9) experiences sudden collapse when we increase
the values of c, α, ˆ
K, and the disease transmission rates, and when we decrease the values of dh, ρ, r, µm.
This is due to the fact that increasing or decreasing the values of these parameters destabilizes the
system and generates fluctuated dynamics. The destabilizing effects generate unstable oscillations. The
amplitudes of oscillations increase until they touch the stable manifold of the extinction equilibrium,
which cause the collapse of the whole colony. The destabilizing effects of c, α, ˆ
K, dmcan be explained
through the dynamics of the virus free subsystem (10) that we will investigate in the next subsection.
•Decreasing the values of c, α, ˆ
Kcan stabilize the system; small values of c, α can cause the extinction
of the infected mite population Si, and lead to the coexistence of Shand Ih.
•Increasing the value of µhcan stabilize the system but large values of µhcan cause extinction of the
whole colony due to the initial oscillations.
•Decreasing µmcan destabilize the system; while increasing it can stabilize the system; large values of
µmcan lead to the extinction of Miand the persistence of Hs, Hi.
•Decreasing the value of ρcould destabilize the system, thus causing the extinction of the colony.
•Increasing the virus transmission rates (i.e., βh,ˆ
βhm,˜
βmh,ˆ
βhm) can stabilize the system, while decreas-
ing their values can destablize the system and cause the extinction of all species.
15
3.3. Dynamics of the disease-free subsystem
Theorem 3.2 in previous section provides sufficient conditions that lead to the following disease-free
dynamics (10)
S′
h=rS2
h
ˆ
K+S2
h−dhSh−αShSm
S′
m=cαShSm−dmSm
whose dynamics can be summarized by the following theorem:
Theorem 3.5 (Dynamics of the disease-free subsystem).Let H∗=dm
αc , M∗=1
αhrH∗
ˆ
K+(H∗)2−dhi.
The disease-free subsystem (10)can have one, three, or four equilibria. The existence and stability conditions
for these equilibria are listed in Table 2. The global dynamics of the disease-free subsystem (10)can be
Equilibria Existence Condition Stability Condition
(0,0) Always exists Always locally stable
(Sc
h,0) r
2√ˆ
K> dhSaddle if Sc
h<dm
αc =H∗; Source if Sc
h>dm
αc =H∗
(S∗
h,0) r
2√ˆ
K> dhSink if S∗
h<dm
αc =H∗; Saddle if S∗
h>dm
αc =H∗
(H∗, M ∗)Sc
h<dm
αc =H∗< S∗
hSink if H∗>pˆ
K; Source if H∗<pˆ
K.
Table 2: The existence and stability of equilibrium for the disease-free subsystem (10), where Sc
h=
r
dh
−rr
dh2−4ˆ
K
2, S∗
h=
r
dh+rr
dh2−4ˆ
K
2and H∗=dm
αc , M∗=1
αhrH ∗
ˆ
K+(H∗)2
−dhi.
summarized as follows:
1. The system (10)converges to extinction (0,0) for almost all initial conditions if r
2√ˆ
K< dhor dm
αc < Sc
h.
2. If S∗
h<dm
αc , depending on initial condition, the trajectory of (10)converges to either (0,0) or (S∗
h,0).
3. If Sc
h<dm
αc < S∗
hthen Model (10)has a unique interior equilibrium (H∗, M∗)which is locally asymp-
totically stable when dm
αc >pˆ
Kand is a source when dm
αc <pˆ
K.
Notes: Theorem 3.5 provides us a global picture on the dynamics of the disease-free subsystem (10), i.e.,
the honeybee colony only infected with mites but not the virus. By applying the results in [57,61], we can
conclude that the disease-free subsystem (10) undergoes a subcritical Hopf-bifurcation at dm
αc =pˆ
K. The
subsystem (10) has a unique unstable limit cycle around (H∗, M ∗) whenever dm
αc <pˆ
K. In this case, the
periodic orbits expand until it touches the stable manifold of the boundary equilibrium (Sc
h,0) which leads
to the extinction of both honeybees and the parasitic mites. We refer to this phenomena as a catastrophic
event which could be linked to CCD. Our theoretical results also suggest that a small death rate for mites
and a large parasitism rate can destabilize the system.
Linking to CCD: To illustrate the catastrophic event, we use reasonable parameters from [56,44]. Let the
reproduction of egg per day during summer be r= 1500; and the population size of the honeybee colony
at which the birth rate is half of the maximum possible rate be pˆ
K= 2000; the natural death rate of
honeybees is dh= 0.01; the parasitism rate is α= 0.005; the energy conversion rate is c= 0.01; and the
natural death rate of mites is dm= 0.1. This set of parameter values gives dm
αc <pˆ
Kwhich implies that
acatastrophic event will occur (see Figure 1; the population of honeybees is in red and collapses around
16
0 20 40 60 80 100 120 140 160 180 200
0
1000
2000
3000
4000
5000
6000
time
Sh
Sm
r = 1500; K= 4000000; dh =0.01;
α=0.005; c = 0.01;dm=0.1;
Sh(0)=H*+1;Sm(0)=50
Figure 1: Time series of Model (10) when r= 1500, α = 0.005, c = 0.01, dh= 0.01, dm= 0.1: population of honeybees is in
red while Varroa mites is in black.
time=200).
Note that ˆ
K=K
ξ2
h
and α= ˆα(1 −ξh)(1 −ξm) where ξh, ξmare ratio of adult bees and ratio of phoretic
stage of Varroa mites in honeybee colony, respectively. The catastrophic event occurs when
dm
αc <pˆ
K⇔dm
ˆα(1 −ξm)c√K<(1 −ξh)
ξh⇔ξh
(1 −ξh)<ˆα(1 −ξm)c√K
dm
which implies that low hive to brood ratio can also destabilize the system and cause the sudden extinction
of honeybees.
3.4. Dynamics of the full system
Recall that the full system (7) of honeybee-mite-virus interactions can be described by the following set
of equations:
S′
h=r(Sh+ρIh)2
ˆ
K+(Sh+ρIh)2−dhSh−βhShIh
Sh+Ih−ˆ
βmhShIm
Sm+Im−˜
βmhShIm−αSh(Sm+Im)
I′
h=ShhβhIh
Sh+Ih+ˆ
βmhIm
Sm+Im+˜
βmhImi−αIh(Sm+Im)−(dh+µh)Ih
S′
m=SmhcαSh−ˆ
βhmIh
Sh+Ih−dmi
I′
m=cα [Ih(Sm+Im) + ShIm] + ˆ
βhmIhSm
Sh+Ih−(dm+µm)Im
.
The results from the previous section provide us a complete picture of the dynamics of the subsystems of
the full system (7). In this subsection, we explore the dynamics of the full system. Here we start with the
17
persistence of honeybees and the extinction of disease and mites. From Theorem 3.1,3.2,3.3,3.4, and 3.5,
we have the following corollary:
Corollary 3.1 (The persistence of honeybees).Assume that r
2√ˆ
K> dh. If N∗<dm
αc and RV
0=βh
dh+µh<
1, the full system (7)converges to the disease-mite-free set DM F ={(Sh, Ih, Sm, Im)∈R4
+:Sm+Ih+Im=
0}where the system (7)is reduced to the following one-D system (14):
S′
h=rS2
h
ˆ
K+S2
h−dhSh(14)
whose dynamics can be summarized as follows:
1. If the inequality r
2√ˆ
K< dhholds, then (14)converges to 0.
2. If the inequalities r
2√ˆ
K> dhand Sh(0) > Sc
hhold, then (14)converges to S∗
h. If the initial condition
falls below Sc
h, i.e., Sh(0) < Sc
h, then (14)also converges to 0.
Moreover, if one of the following inequalities hold
1. r
2√ˆ
K> dh+βh+ˆ
βmh + ( ˜
βmh +α)N∗and Sh(0) >ˆ
Sc
h,or
2. r
2√ˆ
K>max ndh+βh+ˆ
βmh + ( ˜
βmh +α)(N∗−N∗
h),dh+µh+αN∗
ρowith Nh(0) ≥Sh(0) >˜
Sc
h,
then Shis persistent with the following properties:
ˆ
S∗
h≤˜
S∗
h≤lim inf
t→∞ Sh(t)≤lim inf
t→∞ Nh(t)≤lim inf
t→∞ N(t)≤lim sup
t→∞
N(t)≤N∗.
Notes: Based on Theorem 3.1 and Theorem 3.5, we can also conclude that the extinction of disease occurs
when all values of all disease transmission rates, βh,ˆ
βmh,˜
βmh,ˆ
βhm are small; with the consequence that the
full system (7) converges to either (0,0,0,0) or (S∗
h,0,0,0) when RM
0=S∗
h
H∗<1 while (7) converges to either
(0,0,0,0) or (H∗,0, M ∗,0) when 1 <RM
0<S∗
h
Sc
h.
The dynamics of the full system (7) can be extremely complicated. We first look at the existence of
its interior equilibrium. An interior equilibrium (Sh, Ih, Sm, Im) of the system (7), satisfies the following
equations:
0 = SmhcαSh−ˆ
βhmIh
Sh+Ih−dmi
⇒Ih=ShcαSh−dm
dm+ˆ
βhm−cαSh,dm
cα < Sh<dm+ˆ
βhm
cα
⇒Nh=Sh+Ih=ˆ
βhmSh
dm+ˆ
βhm−cαSh=g1(Sh)
(15)
0 = r(Sh+ρIh)2
ˆ
K+(Sh+ρIh)2−dh(Sh+Ih)−α(Sh+Ih)(Sm+Im)−µhIh
⇒Nm=Sm+Im=
r(Sh+ρIh)2
ˆ
K+(Sh+ρIh)2−dh(Sh+Ih)−µhIh
α(Sh+Ih)
⇒Nm=rSh(ˆ
βhm+(dm−cαSh))( ˆ
βhm+(dm−cαSh)(1−ρ))2
αˆ
βhmhˆ
K(ˆ
βhm+(dm−cαSh))2+S2
h(ˆ
βhm+(dm−cαSh)(1−ρ))2i−dh
α−µh(cαSh−dm)
ˆ
βhm =g2(Sh)
(16)
18
0 = cα(Sh+Ih)(Sm+Im)−dm(Sm+Im)−µmIm
⇒Im=Nm(cα(Sh+Ih)−dm)
µm=Nm[cα ˆ
βhmSh−dm(ˆ
βhm+(dm−cαSh))]
µm(ˆ
βhm+(dm−cαSh))=Nm(cαSh