A MATHEMATICAL MODEL FOR THE SPATIAL ...

A MATHEMATICAL MODEL FOR THE SPATIAL SPREAD AND BIOCONTROL OF THE ASIAN LONGHORNED BEETLE STEPHEN A. GOURLEY

† AND

YIJUN LOU

‡

Abstract. We propose a mathematical model, of four coupled delay differential equations, for control of the Asian longhorned beetle Anoplophora glabripennis by one of its natural predators, the cylindrical bark beetle Dastarcus longulus or another predator with similar characteristics. It is a predator prey interaction at the larval rather than adult level which creates interesting modelling challenges. We specify the birth rate only for A. glabripennis and calculate the birth rate of the control agent D. longulus by keeping track of its consumption of larval A. glabripennis biomass and using the idea of conversion of biomass. We prove rigorous results on the stability of equilibria and on persistence of D. longulus and we make an assessment of the kinds of characteristics that enable D. longulus, or any similar control agent, to effectively control A. glabripennis. A pest such as A. glabripennis will destroy its habitat and must continually find new host trees. Even though our model does not have explicit spatial dependence, we may use it to make some inferences about the likely spatial spread of an infestation. Key words. Asian longhorned beetle, biocontrol, delay, age-structure, persistence, stability AMS subject classifications. 34K25, 34K60, 92D40

1. Introduction. The Asian longhorned beetle (ALB) is a wood-boring beetle capable of killing healthy trees. It is native to Asia, including China and Korea, and was intercepted in international trade in 1992, mostly in wood packaging material. It is non-indigenous in USA and was first detected in Brooklyn, NY, in 1996. Now, it is established in North America and Europe [7, 9]. The ALB can potentially infest many deciduous tree species, particularly maples which comprise about 30% of all urban trees in the eastern USA, with the potential for massive economic loss if the pest is allowed to spread, whether by natural dispersal or incidental transport (Smith [20]). Other trees that can host ALB include willow, elm, birch, ash and poplar. The total ALB life cycle from egg to beetle lasts one to two years in Asia, North America, central and southern Europe. Most of the time is taken up within the larval stage during which larvae tunnel deep into a tree’s heartwood. As they tunnel they feed on the phloem and cambium layers beneath the bark, and later the sapwood and heartwood. They pupate on reaching a critical mass, and the adults emerge from the tree through a circular exit hole. Adults emerge from spring onwards and live a few months at most [7]. Currently, the only widely used method of control of the ALB is the identification and removal of infested trees. However, identification is difficult because for most of its life cycle the ALB is hidden deep within the wood of the tree and the relatively short adult lifespan is spent mainly in the upper canopy. Difficulties of detection were both discussed and modelled in an earlier paper by Gourley and Zou [6]. However, the ALB has a few natural predators. Apart from the cylindrical bark beetle (CBB) Dastarcus longulus [7, 13], these include clerid beetles, some flies including robber flies, several wasps, some nematodes and fungi, some birds (particularly woodpeckers) and some small mammals [21]. In view of the difficulties involved in the detection and removal of infested trees, the possibility of effective control of the ALB using a natural predator is of considerable interest. There are very few articles in scientific journals on the CBB Dastarcus longulus as a † Corresponding author. Department of Mathematics, University of Surrey, Guildford, Surrey, GU2 7XH, UK. E-mail: [email protected]. Fax: +44 1483 686071. ‡ Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong. This author’s research was partially supported by grants from Hong Kong Polytechnic University.

1

2

S.A. Gourley and Y. Lou

control agent for the ALB, or even on control of the ALB using other natural predators. However, biocontrol using the CBB and other agents is discussed in the work of Smith and coauthors [21, 22, 23, 30] and it is evident from articles in USA governmental websites that biocontrol using a natural predator is seen as a possible way forward. However, we do not know the likely behaviour of the CBB in North America and there is the risk that it might prefer to attack indigenous species rather than the larvae of the ALB. The difficulties involved in carrying out the necessary experimentation no doubt explain the lack of available information on the matter. However, we do know that in China the CBB is an efficient larval/pupal parasitoid of ALB. It has been found to parasitize and kill as much as 60% of ALB. The first instar larvae of CBB have thoracic legs and crawl in search of a suitable host ALB larva [9, 28]. On finding such a host, the CBB larva loses its legs and attaches to the host’s body. As many as thirty CBB individuals may complete their development on a single ALB larva or pupa, killing it within ten days [20]. In this paper we study the potential for effective biocontrol of the Asian longhorned beetle (ALB) Anoplophora glabripennis by one of its natural predators, the cylindrical bark beetle (CBB) Dastarcus longulus. For that purpose, we derive a model consisting of four differential equations that describe the evolution of the populations of larval and adult CBB and ALB. Although two species are involved, we prescribe the birth rate function for only one of them, the ALB. For the CBB, we model the fact that their larvae consume larval ALB biomass as they develop, and when they mature this consumed larval ALB biomass is converted into CBB biomass in the form of CBB eggs. This enables us to compute the CBB egg laying rate, rather than simply specifying it. Furthermore, our modelling accounts for conversion efficiency and loss of some consumed larval ALB biomass due to the failure of some CBB to mature. Although the model we propose does not explicitly involve spatial effects, we will still use it to make inferences about the likely spatial spread of an ALB infestation in a forested area. For example, we know that an adult might fly up to a kilometer or two in search of a new host tree, but would not normally do so if the tree in which it matured is still suitable for egg laying. That tree will eventually become unsuitable, however, due to accumulated damage from many generations of beetles. The tunnels formed by the ALB larvae cause permanent damage to the tree including structural weakness and disruption of the tree’s vascular system [7]. Since it is the accumulated damage to a tree over many beetle generations that triggers dispersal to other trees, we believe that simple representations of dispersal, such as the use of reactiondiffusion equations based on random walk arguments, are not appropriate. Instead, since adult ALB do not seek a new host tree unless they have to, we consider the situation when the infested zone is maximally infested so that emerging adults must seek new host trees, and then simply describe their movements by how much they must expand the infested zone in order to acquire enough new host trees to accommodate the eggs they will lay. This approach enables us to estimate the rate of spread of the infestation in terms of parameters that are measurable. Various models have previously been proposed to investigate the impact of biological control strategies in various scenarios. These include the introduction of a competitor snail species to control schistosomiasis [1], disease introduction to invasive predators to preserve prey [4, 18] and the introduction of larvivorous fish to control malaria [14]. Several models have been formulated to measure the effects of biological agents against pests and these include the sterile male technique [12], and the introduction of natural enemies or parasitoids [5, 8, 15, 16, 25]. However, our modelling process is different because of the particular nature of the interaction (immature prey being consumed by immature predators). This necessitates particular care in the incorporation of the age structure of both pest and predator and the modelling of the consumption of biomass.

3

Biocontrol of an invasive beetle

?hmodelderivi?

2. Model derivation. Throughout this manuscript, the subscripts alb and cbb stand for Asian longhorned beetle (ALB) Anoplophora glabripennis and cylindrical bark beetle (CBB) D. longulus, respectively. Some variables such as the per-capita mortality rates also have superscripts l or a, which stand respectively for larva and adult. First, we derive an equation for the number Lalb (t) of larval Asian longhorned beetles at time t. We introduce lalb (t, a) to denote the density of ALB larvae at time t of age a. These larvae are preyed upon by the larvae of CBB. Using a standard von-Foerster age structured modelling approach, we write ∂lalb (t, a) ∂lalb (t, a) eσlalb (t, a)Lcbb (t) + = −µlalb lalb (t, a) − ∂t ∂a 1 + heσLalb (t)

(2.1) ?080413_1?

to describe the loss of ALB larvae either through natural death with per-capita mortality rate µlalb , or through predation by CBB larvae as described by the last term in (2.1) using a Holling type II functional response, with Lcbb (t) denoting the number of CBB larvae at time t. The way this works becomes clearer if one looks further down at equation (2.5), in which the corresponding term has Lalb (t) in both its numerator and denominator and therefore levels off at large values of Lalb (t), thereby modelling the fact that each CBB larva can only consume a limited quantity of ALB larval biomass per unit time, however much is available. This is especially important in view of the observation that a single ALB larva offers rather a lot of food to an individual CBB larva. The quantity e stands for the ALB larval biomass encounter rate (the exponential function is denoted exp throughout, to avoid confusion), h is the handling (digestion) time per unit biomass consumed and σ is the fraction of encountered ALB larval biomass that is actually eaten. The total number of ALB larvae at time t is Z τalb lalb (t, a) da Lalb (t) = 0

where τalb is the maturation time for ALB. In fact ALB larvae do not pupate until they have reached a critical mass and the time taken to do so depends on the weather. In view of the difficulties of modelling this dependence, we still treat τalb as a known constant here. Differentiating, and using (2.1), we have dLalb (t) eσLalb (t)Lcbb (t) = lalb (t, 0) − lalb (t, τalb ) − µlalb Lalb (t) − . dt 1 + heσLalb (t)

(2.2) ?080413_4?

Now, lalb (t, 0) is the egg laying rate of ALB; this is taken to be a function b(·) of the total number Aalb (t) of adult ALB, on the assumption that those beetles remain reproductively active throughout the great majority of their relatively short adult lives, usually considered a reasonable assumption for modelling insect populations. Thus lalb (t, 0) = b(Aalb (t)).

(2.3) ?080413_2?

ξ Next, we calculate lalb (t, τalb ). To do so, define lalb (a) = lalb (a + ξ, a). Differentiating, and using (2.1), we get ξ ξ dlalb (a) eσlalb (a)Lcbb(a + ξ) ξ = −µlalb lalb (a) − da 1 + heσLalb (a + ξ)

so that ξ lalb (a)

=

ξ lalb (0) exp

Z −

0

a

µlalb +

eσLcbb(η + ξ) 1 + heσLalb (η + ξ)

dη .

4


Setting a = τalb and ξ = t − τalb and using (2.3), we obtain Z τalb eσLcbb (η + t − τalb ) l µalb + lalb (t, τalb ) = b(Aalb (t−τalb)) exp − dη . 1 + heσLalb (η + t − τalb ) 0 (2.4) ?080413_3? Inserting (2.3) and (2.4) into (2.2) gives us the larval ALB equation: dLalb (t) eσLalb (t)Lcbb (t) = −µlalb Lalb (t) − + b(Aalb (t)) dt 1 + heσLalb (t) Z t eσLcbb (η) dη , µlalb + − b(Aalb (t − τalb ))exp − 1 + heσLalb (η) t−τalb

(2.5) ?L_alb?

which can also be written in the integral equation form Z t Z t eσLcbb (η) dη dξ. µlalb + Lalb (t) = b(Aalb (ξ)) exp − 1 + heσLalb (η) t−τalb ξ

(2.6) ?L_alb_int?

The last term in (2.5) is the rate at which larval ALB mature into adult beetles. Thus, we may immediately write down an equation for the number Aalb (t) of adult ALB: dAalb (t) dt

=

−µaalb Aalb (t)

+

Z b(Aalb (t − τalb ))exp −

t

t−τalb

µlalb +

eσLcbb(η) 1 + heσLalb (η)

dη ,

(2.7) ?A_alb? where µaalb is the per-capita mortality rate for adult ALB. Next, we need equations for the larval and adult CBB. The second term in the right hand side of (2.5) describes predation of larval ALB by larval CBB. The consumed larval ALB biomass is converted into CBB biomass, as with any predatorprey interaction, but in this case the process is obviously not instantaneous. We compute this conversion as follows. Consider the CBB larvae that mature at time t. Soon after maturation they mate and lay eggs, and this is assumed to occur with only negligible delay. Let τcbb be the larval developmental time for CBB. Of those CBB larvae that mature at time t, between times ξ and ξ + dξ with ξ ∈ (t − τcbb , t) these larvae consumed a quantity exp(−µlcbb (t − ξ))

eσLalb (ξ)Lcbb (ξ) dξ 1 + heσLalb (ξ)

of larval ALB biomass. The exponential term exp(−µlcbb (t − ξ)) corrects for CBB larval mortality over the time interval [ξ, t]. Without it we would be computing all larval ALB biomass consumed over the time interval [ξ, ξ + dξ] including consumption by those CBB larvae that have since died. The total amount of larval ALB biomass consumed by those CBB larvae that have survived and are ready to mature at time t is Z t eσLalb (ξ)Lcbb (ξ) exp(−µlcbb (t − ξ)) dξ. 1 + heσLalb (ξ) t−τcbb At time t (or, in practice, soon after) this larval ALB biomass is converted into new CBB biomass in the form of a number of CBB eggs. If the above quantity is multiplied by a factor B, which measures the efficiency of this conversion, we obtain a number of eggs laid by those CBB that have just matured at time t. This number has to be converted into a rate (the egg-laying rate). The eggs might in practice be

5


laid over a very short time interval after maturation, or maybe even all at once, but they are attributable to a cohort of beetles that have taken a time τcbb to mature. Therefore we divide by τcbb and it may be said that the CBB have an egg laying rate lcbb (t, 0) given by Z t B eσLalb (ξ)Lcbb (ξ) lcbb (t, 0) = dξ (2.8) ?080413_5? exp(−µlcbb (t − ξ)) τcbb t−τcbb 1 + heσLalb (ξ) where lcbb (t, a) is the age density function for CBB. Since τcbb is the larval developmental time for CBB, ∂lcbb(t, a) ∂lcbb(t, a) + = −µlcbb lcbb (t, a), 0 < a < τcbb . ∂t ∂a Integrating along characteristics leads to Z eσLalb (ξ)Lcbb (ξ) exp(−µlcbb τcbb )B t−τcbb dξ exp(−µlcbb (t−τcbb −ξ)) lcbb (t, τcbb ) = τcbb 1 + heσLalb (ξ) t−2τcbb (2.9) ?080413_6? for t > 2τcbb. Expression (2.9) states that the CBB maturation rate at time t is the CBB larval survival probability exp(−µlcbbτcbb ) multiplied by the birth rate lcbb (t − τcbb , 0) at the earlier time t − τcbb . The latter depends on the quantity of larval ALB biomass consumed by the previous generation over the earlier time interval [t − 2τcbb , t − τcbb ]. Having noted this interpretation of (2.9), we combine and simplify the exponentials therein. Now Z τcbb lcbb(t, a) da, Lcbb (t) = 0

differentiation of which leads to dLcbb (t) dt

= −µlcbb Lcbb(t) + −

B τcbb B τcbb

Z Z

t t−τcbb

exp(−µlcbb (t − ξ))


t−τcbb


t−2τcbb

(2.10) ?L_cbb?


by using (2.8)–(2.9). There is the following integral equation as an alternative to (2.10): Z t Z η B eσLalb (ξ)Lcbb (ξ) Lcbb(t) = dξ dη. (2.11) ?L_cbb_int? exp(−µlcbb (t − ξ)) τcbb t−τcbb η−τcbb 1 + heσLalb (ξ) For the number Acbb (t) of adult CBB at time t, we have the following equation dAcbb (t) dt

= +

−µacbbAcbb (t) B τcbb

Z

t−τcbb

t−2τcbb

exp(−µlcbb (t

eσLalb (ξ)Lcbb (ξ) dξ. − ξ)) 1 + heσLalb (ξ)

(2.12) ?A_cbb?

The system to be solved comes in two versions. One consists of equations (2.5), (2.7), (2.10) and (2.12). The other consists of equations (2.6), (2.7), (2.11) and (2.12). The two systems are equivalent for the restricted class of initial data such that (2.6) and (2.11) hold at time t = 0. Since these two integral equations come about through the modelling process, and have a clear ecological interpretation, the need for them to hold at t = 0 is not considered a restriction at all. Any ecologically realistic initial data must satisfy those conditions.

6


3. Model analysis. Rigorous analysis of either version of the model of Section 2 is not easy for a general birth function b(·) of ALB. We make the following biologically acceptable assumptions on the ALB egg laying rate b(·): b(0) = 0, b(A) is increasing and b0 (A) is decreasing for A > 0, there exists l a ∗∗ A∗∗ alb > 0 such that exp(−µalb τalb )b(A) > µalb A when 0 < A < Aalb , and l a ∗∗ exp(−µalb τalb )b(A) < µalb A when A > Aalb . (3.1) ?110413_3? Note that (3.1) implies b0 (0) exp(−µlalb τalb ) > µaalb .

(3.2) ?090413_4?

The fact that b0 (A) is decreasing assures us that the inequality in (3.2) is indeed strict. 3.1. Positivity and boundedness. We first prove the positivity and boundedness of solutions for the model system. Proposition 3.1. Suppose that (3.1) holds, that all four variables are ?hpos_boundedi? non-negative and continuous on their respective initial intervals, and that (2.6) and (2.11) hold at t = 0. Then all components of the solution of system (2.5), (2.7), (2.10) and (2.12), or of the variant system (2.6), (2.7), (2.11) and (2.12), are non-negative and bounded for all t ≥ 0. Proof. We first show that the variables Lalb , Aalb , Lcbb and Acbb , with nonnegative initial data, remain nonnegative as long as they exist. We start by establishing the nonnegativity of Aalb by using [19, Theorem 5.2.1]. Since if Lalb , Lcbb ≥ 0 initially and Aalb (0) = 0, then the right hand side of (2.7) is nonnegative. Therefore, Aalb (t) ≥ 0. Similarly, Theorem 5.2.1 in [19] also implies the nonnegativity of Acbb . The nonnegativity of Lalb and Lcbb follow from the integral equations (2.6) and (2.11). Next we show that Aalb (t) is bounded. Let r > 0 be arbitrary. Then, on the interval [−τalb , r], Aalb (t) assumes its maximum at some value tm . If tm ∈ [−τalb , 0] then obviously Aalb (t) ≤ maxθ∈[−τalb,0] Aalb (θ) for all t ∈ [−τalb , r]. Suppose that tm ∈ (0, r]. Then A0alb (tm ) ≥ 0 and Aalb (tm ) ≥ Aalb (tm − τalb ). Therefore, from (2.7), we obtain 0 ≤ −µaalb Aalb (tm ) + exp(−µlalb τalb )b(Aalb (tm − τalb )) ≤ −µaalb Aalb (tm ) + exp(−µlalb τalb )b(Aalb (tm ))

since b is increasing. It follows from (3.1) that Aalb (tm ) ≤ A∗∗ alb and therefore Aalb (t) ≤ A∗∗ for all t ∈ [−τ , r]. Since r > 0 is arbitrary, it follows that alb alb Aalb (t) ≤ max

max

θ∈[−τalb ,0]

Aalb (θ),

A∗∗ alb

for all t ≥ −τalb

which establishes the boundedness of Aalb (t). We now show that A∗∗ alb is the asymptotic bound. By the fluctuation method (Thieme [27]), there is a sequence of times ti such that ti → ∞, Aalb (ti ) → Aalb = lim supt→∞ Aalb (t) and A0alb (ti ) → 0 as i → ∞. From (2.7), it follows that A0alb (ti ) ≤ −µaalb Aalb (ti ) + exp(−µlalb τalb )b(Aalb (ti − τalb )). Letting i → ∞, applying standard properties of the lim sup and using the assumption that b(·) is increasing, we get 0 ≤ −µaalb Aalb + exp(−µlalb τalb )b(Aalb ).

7


From (3.1), lim supt→∞ Aalb (t) = Aalb ≤ A∗∗ alb . Next, from (2.6) we can show that lim sup Lalb (t) ≤ t→∞

l b(Aalb )(1 − exp(−µlalb τalb )) b(A∗∗ alb )(1 − exp(−µalb τalb )) ≤ = L∗∗ alb . µlalb µlalb

Finally, we show that Lcbb and Acbb are bounded. From (2.5), dLalb (t) eσLalb (t)Lcbb (t) = b(Aalb (t)) − − µlalb Lalb (t) 1 + heσLalb (t) dt Z t eσLcbb (η) l dη − b(Aalb (t − τalb ))exp − µalb + 1 + heσLalb (η) t−τalb ≤ b(Aalb (t)) −

dLalb (t) . dt

Therefore, for any t ≥ η, we have Z η eσLalb (ξ)Lcbb (ξ) dξ exp −µlcbb(t − ξ) 1 + heσLalb (ξ) η−τcbb Z η ≤ b(Aalb (ξ))dξ − (Lalb (η) − Lalb (η − τcbb )) η−τcbb

≤ τcbb

sup

η−τcbb ≤ξ≤η

{b(Aalb (ξ))} + Lalb (η − τcbb ).

Using the integral equation (2.11) for Lcbb, it follows that " # Z t B Lcbb (t) ≤ τcbb sup {b(Aalb (ξ))} + Lalb (η − τcbb ) dη. τcbb t−τcbb η−τcbb ≤ξ≤η Since Aalb and Lalb are bounded, Lcbb is also bounded. From the boundedness of Lcbb , it is easy to check that Acbb is bounded by using (2.12) and a comparison argument. 3.2. Existence of equilibria. Biological insights can be gained from a study of the equilibria of the model. In this section, we establish the existence of equilibria. 3.2.1. CBB-free equilibrium. If (3.1) holds there is an equilibrium with Lcbb = Acbb = 0, which we refer to as the CBB-free equilibrium. In this equilibrium, ∗∗ ∗∗ the Lalb and Aalb components are L∗∗ alb and Aalb , with Aalb > 0 from (3.1), which satisfy a ∗∗ exp(−µlalb τalb )b(A∗∗ alb ) = µalb Aalb ,

?hequilCBBpresenti?

L∗∗ alb =

l b(A∗∗ alb )(1 − exp(−µalb τalb )) . l µalb

(3.3) ?110413_1?

3.2.2. Equilibrium with the CBB present. Since the CBB is seen as a control agent for ALB, we are particularly interested in the possible existence of an equilibrium in which all four components are positive but the ALB larvae and adults are present only in smaller numbers. Equation (2.10) shows that, in any such equilibrium (L∗alb , A∗alb , L∗cbb, A∗cbb ) with L∗cbb > 0, we must have L∗alb =

τcbb (µlcbb )2

2.

Beσ(1 − exp(−µlcbb τcbb ))2 − heστcbb (µlcbb )

Observe that, necessarily, 2

B(1 − exp(−µlcbb τcbb ))2 > hτcbb (µlcbb ) .

(3.4) ?L_alb_equil?

8


Expression (3.4) is never singular in practice. This is because, as we shall show, a stronger condition on B is in fact necessary for the existence of an equilibrium with the CBB present. This is condition (3.7), which also features later as one of the conditions for persistence of the CBB. With L∗alb given by (3.4), the L∗cbb and A∗alb components of an equilibrium with the CBB present are found by simultaneously solving eσL∗cbb b(A∗alb ) ∗ l (3.5) ?090413_2? Lalb = 1 − exp −τalb µalb + eσL∗cbb 1 + heσL∗alb µlalb + 1 + heσL∗alb and µaalb A∗alb = b(A∗alb ) exp −τalb µlalb +

eσL∗cbb 1 + heσL∗alb

.

(3.6) ?090413_3?

The following result deals with the existence of L∗cbb > 0 and A∗alb > 0 satisfying (3.5) and (3.6). Later, we obtain more explicit expressions for the equilibrium components in the case when B is very large. Proposition 3.2. If assumption (3.1) holds, together with ?hCBBpresentequilibriumi? 2 l BeσL∗∗ alb 1 − exp −µcbb τcbb > (µlcbb )2 , (3.7) ?120413_5a? τcbb (1 + heσL∗∗ alb ) then system (2.5), (2.7), (2.10) and (2.12) has an equilibrium in which the ALB and CBB coexist. Proof. Each of (3.5) and (3.6) defines a curve in the (L∗cbb , A∗alb ) plane (recall that L∗alb is fixed and given by (3.4)). We can write equation (3.5) in the form A∗alb = φ1 (L∗cbb ) where   ∗ L o  n alb φ1 (x) = b−1  eσx l τalb k τalb µalb + 1+heσL ∗ alb

with k(x) = (1 − exp(−x))/x. Now, when x > 0, k(x) is decreasing in x and b−1 (·) is increasing, since b is. Thus, φ1 (x) is increasing. Now consider the curve defined by (3.6). For a particular L∗cbb, let the function φ2 (L∗cbb ) be defined as the solution A∗alb of (3.6) so that (3.6) is rewritten as A∗alb = φ2 (L∗cbb ). It is easy to see, using (3.1), that the function φ2 (x) thus defined is decreasing. This is because if we increase the value of L∗cbb we decrease the coef∗ a ficient of the function n b(·)in (3.6). The value o Aalb at which the curves A → µalb A eσL∗

cbb and A → b(A) exp −τalb µlalb + 1+heσL intersect must therefore decrease as ∗ alb ∗ we increase Lcbb. Moreover, this value must reach zero at a finite value of L∗cbb. In view of these facts about the curves A∗alb = φ1 (L∗cbb ) and A∗alb = φ2 (L∗cbb ) in the (L∗cbb , A∗alb ) plane, it follows immediately that if φ1 (0) < φ2 (0) then there exist L∗cbb > 0 and A∗alb > 0 satisfying (3.5) and (3.6). Now L∗alb µlalb L∗alb −1 −1 φ1 (0) = b =b τalb k(τalb µlalb ) 1 − exp(−µlalb τalb )

while φ2 (0) = A∗∗ alb . Thus, the condition φ1 (0) < φ2 (0) becomes b(A∗∗ alb ) >

µlalb L∗alb . 1 − exp(−µlalb τalb )

This can be shown to be equivalent to (3.7), using (3.4) and the expression for L∗∗ alb in (3.3).


9

Remark 3.1. The proof of Proposition 3.2 also implies that the coexistence ∗ equilibrium (L∗alb , A∗alb , L∗cbb , A∗cbb ) is unique. Moreover, L∗alb < L∗∗ alb and Aalb < ∗∗ Aalb if the coexistence equilibrium exists. If the assumptions of Proposition 3.2 hold, further useful insight can be gained by supposing the parameter values are such that the equilibrium ALB larval population is low, i.e. expression (3.4) is low. For example, we might assume that the conversion efficiency B is large. Then we also expect the equilibrium adult ALB population A∗alb to be on the low side, so that b(A∗alb ) ≈ b0 (0)A∗alb . From (3.6) we then obtain an explicit expression for L∗cbb: 0 1 b (0) 1 + heσL∗alb l − µ (3.8) ?L_cbb_equil? ln L∗cbb = alb eσ τalb µaalb with L∗alb given by (3.4). Then, from (3.5), A∗alb =

L∗alb ln(b0 (0)/µaalb ) . b0 (0)τalb (1 − µaalb /b0 (0))

(3.9) ?A_alb_equil?

Expression (3.9) is automatically positive, while expression (3.8) is positive because of (3.2). From (2.12), A∗cbb is given in terms of other equilibrium components by A∗cbb =

Beσ exp(−µlcbb τcbb )(1 − exp(−µlcbb τcbb ))L∗alb L∗cbb µlcbb µacbb τcbb (1 + heσL∗alb )

(3.10) ?A_cbb_equil?

without further parameter restrictions. We conclude that if the numbers of adult and larval ALB are small (for example, if B is very large) then the equilibrium in which the CBB are present is given approximately by (3.4), (3.9), (3.8) and (3.10). 3.3. Global stability of the CBB-free equilibrium. The following theorem shows that if the biomass conversion efficiency B is too low, or if the CBB larvae take too long to digest their food (h is too large) then the CBB cannot control the ALB and will be driven to extinction. The same outcome holds if the CBB mature too quickly (τcbb is low), since in that case they fail to consume enough ALB larval biomass. Theorem 3.1. Suppose that ?hglobstabCBBfreei? 2 l BeσL∗∗ alb < (µlcbb )2 (3.11) ?110413_2? 1 − exp −µ τ cbb cbb τcbb (1 + heσL∗∗ ) alb with L∗∗ alb given in (3.3), and that (3.1) holds. Then the CBB-free equilibrium ∗∗ (Lalb , Aalb , Lcbb , Acbb ) = (L∗∗ alb , Aalb , 0, 0) of system (2.5), (2.7), (2.10) and (2.12) is globally asymptotically stable for all non-negative solutions such that Aalb (θ) 6≡ 0 on [−τalb , 0] and such that (2.6) and (2.11) hold at t = 0. Proof. We use the variant of the model that involves the integral equations (2.6) and (2.11). From (2.7), and using positivity of solutions, dAalb (t) ≤ −µaalb Aalb (t) + exp(−µlalb τalb )b(Aalb (t − τalb )). dt

(3.12) ?110413_6?

Since b(·) is increasing, we may use a comparison argument (see, for example Smith [19]) to conclude that Aalb (t) is bounded by the solution of the corresponding differential equation obtained from (3.12) by changing ≤ to =. Since b(·) is increasing, positive solutions of that differential equation approach A∗∗ alb (see Kuang [10]). Therefore, lim sup Aalb (t) ≤ A∗∗ alb t→∞

10


and, from (2.6), lim sup Lalb (t) ≤ t→∞

l b(A∗∗ alb )(1 − exp(−µalb τalb )) = L∗∗ alb . µlalb

Since (3.11) holds, there exists > 0 such that (µlcbb )2

2 l Beσ(L∗∗ alb + ) . > 1 − exp −µcbb τcbb τcbb (1 + heσ(L∗∗ alb + ))

(3.13) ?120413_1?

With this , Lalb (t) ≤ L∗∗ alb + for t sufficiently large. From (2.11) we obtain, for t sufficiently large, Lcbb(t) ≤

B τcbb

Z

t

Z

η

t−τcbb η−τcbb


eσ(L∗∗ alb + )Lcbb (ξ) dξ dη (3.14) ?110413_4? 1 + heσ(L∗∗ alb + )

because the integrand of (2.11) increases with respect to Lalb (ξ). Any solution ˜ cbb (t) of the corresponding of inequality (3.14) is bounded above by a solution L integral equation ˜ cbb(t) = B L τcbb

Z

t

Z

η

t−τcbb η−τcbb


˜ eσ(L∗∗ alb + )Lcbb (ξ) dξ dη (3.15) ?Eq:ComSyst? ∗∗ 1 + heσ(Lalb + )

˜ cbb (θ) ≥ Lcbb(θ) for all θ ∈ [−2τcbb , 0]. Indeed, it is straightforward to such that L ˜ cbb(t) − show, since (3.15) is linear and has a positive kernel, that the variable L Lcbb (t) can never go negative. Actually, (3.14) only holds for t sufficiently large but ˜ cbb(t) the comparison still holds after a translate in time, for a suitable solution L ˜ cbb(θ) ≥ 0 for of (3.15). In fact the solution map of (3.15) is strongly positive: if L ˜ cbb(θ0 ) > 0 for some θ0 ∈ [−2τcbb, 0], then L ˜ cbb(t) > 0 for all θ ∈ [−2τcbb, 0] and L all t > 2τcbb . Moreover, since (3.15) has a positive kernel, by the Krein-Rutman theorem it suffices to consider only the real roots of the characteristic equation that ˜ cbb (t) = exp(λt), which is results from a search for solutions of the form L (λ +

µlcbb )2

2 Beσ(L∗∗ l alb + ) . 1 − exp −(λ + µcbb )τcbb = τcbb (1 + heσ(L∗∗ alb + ))

(3.16) ?110413_5?

˜ cbb(t) → 0 (and hence Lcbb (t) → 0) as In view of all these facts, to show that L t → ∞, it suffices to prove that all the real roots of (3.16) are negative. Note that, since (3.13) holds, the left hand side of (3.16) exceeds its right hand side when λ = 0. For notational simplicity, we denote A=

Beσ(L∗∗ alb + ) . τcbb (1 + heσ(L∗∗ alb + ))

2 l < (µlcbb )2 . The real roots of (3.16) must Then we have A 1 − exp −µcbbτcbb satisfy one of the following equations: √ f (λ) := √A(1 − exp −(λ + µlcbb )τcbb ) + (λ + µlcbb ) = 0, A(1 − exp −(λ + µlcbb )τcbb ) − (λ + µlcbb ) = 0. g(λ) :=

The function y = f (λ) satisfies lim f (λ) = −∞, lim f (λ) = ∞ and f 0 (λ) > 0. λ→∞ √ λ→−∞ Moreover, f (0) = A(1 − exp −µlcbb τcbb ) + µlcbb > 0. Therefore, the equation f (λ) = 0 has just one real root and it is negative.

11


The function y = g(λ) satisfies lim g(λ) = −∞, lim g(λ) = −∞. Moreover, λ→−∞ λ→∞ √ √ g (λ) = Aτcbb exp{−(λ + µlcbb√)τcbb } − 1. Let g 0 (λ0 ) = 0, then Aτcbb exp{−(λ0 + 0

A) µlcbb )τcbb } = 1 and λ0 = ln(ττcbb − µlcbb. It is easy to see that g 0 (λ) > 0 if λ < λ0 cbb √ and g 0 (λ) < 0 if λ > λ0 . Since g(0) = A(1 − exp −µlcbb τcbb ) − µlcbb < 0, to show that the equation g(λ) = 0 has no positive real roots, it suffices to prove that √ 2 l 0 < (µlcbb )2 , we have A < g (0) < 0. In fact, since A 1 − exp −µcbb τcbb µlcbb . 1−exp{−µlcbb τcbb }

g 0 (0) = < =

Therefore,

√ Aτcbb exp(−µlcbbτcbb ) − 1

µlcbb τ exp(−µlcbb τcbb ) − 1 1−exp{−µlcbb τcbb } cbb 1 [µl τ exp(−µlcbbτcbb ) 1−exp{−µlcbb τcbb } cbb cbb

− (1 − exp(−µlcbb τcbb ))].

Define h(u) := u exp(−u) − 1 + exp(−u). It is easy to show that h(u) < 0 for u > 0, and therefore g 0 (0) < 0. Hence, the equation g(λ) = 0 has no real positive roots. Therefore, all real roots of (3.16) are negative and hence Lcbb (t) → 0 as t → ∞. From (2.12) it follows that Acbb (t) → 0. Equation (2.7) can be considered as an asymptotically autonomous equation with the following limit equation: dAalb (t) = −µaalb Aalb (t) + exp(−µlalb τalb )b(Aalb (t − τalb )), dt the solution of which tends to A∗∗ alb as noted earlier in this proof, since Aalb (θ) 6≡ 0 on [−τalb , 0]. This argument can be justified using established theories on asymptotically autonomous systems; see for example Mischaikow, Smith and Thieme [17] and Thieme [26]. Finally, the integral equation (2.6), in the limit when t → ∞, shows that Lalb (t) → L∗∗ alb . The proof is complete. 3.4. Persistence of CBB. We show under condition (3.17) that the CBB uniformly and strongly persist. Uniform strong persistence of CBB implies that the CBB does not go extinct and also that, except for initial transients, their numbers will always be above some minimum threshold that does not depend on the initial conditions. Moreover, the CBB and ALB will coexist. Theorem 3.3. Suppose that (3.1) holds and that ?hweakpersisti?

2 l BeσL∗∗ alb > (µlcbb )2 1 − exp −µ τ cbb cbb τcbb (1 + heσL∗∗ ) alb

(3.17) ?120413_5?

with L∗∗ alb given in (3.3). Then the CBB and ALB uniformly persist in the sense that there exists η > 0 such that lim inf xi (t) > η, i = 1, 2, 3, 4 t→∞

for all solutions x(t) = (Lalb , Aalb , Lcbb, Acbb ) of (2.5), (2.7), (2.10) and (2.12) such that (2.6) and (2.11) hold at t = 0, and Aalb (θ) 6≡ 0 on [−τalb , 0]. Proof. Denote τ = max{τalb , 2τcbb },

M := C([−τ, 0], R4+ ),

M0 := {φ ∈ M : φi (0) > 0, i = 1, 2, 3, 4} and ∂M0 := M \M0 . Clearly, M0 is an open set relative to M . Define the solution semiflow Φ(t) by Φ(t)φ(θ) = (Lalb (t + θ), Aalb (t + θ), Lcbb (t + θ), Acbb (t + θ)),

12


where (Lalb (t), Aalb (t), Lcbb (t), Acbb (t)) is the solution of the model system with initial data φ. It then follows from Proposition 3.1 that Φ(t) is point dissipative and Φ(t)M0 ⊂ M0 . Let ω(φ) be the omega limit set of the orbit γ + (φ) := {Φ(t)φ : ∀t ≥ 0} and define M∂ := {φ ∈ ∂M0 : Φ(t)φ ∈ ∂M0 , t ≥ 0}. In view of the proof of Theorem 3.1, we have ∗∗ ω(φ) = {(0, 0, 0, 0), (L∗∗ alb, Aalb , 0, 0)},

∀φ ∈ M∂ .

Now, we prove the following claim. Claim: There exists > 0 such that, for all solutions (Lalb , Aalb , Lcbb , Acbb ) with Aalb (θ) 6≡ 0 on [−τalb , 0], we have lim sup Lcbb (t) ≥ . t→∞

Suppose the claim is false. Then for any > 0 there is a solution with Lcbb <

(3.18) ?120413_7?

where the overbar is defined by Lcbb = lim supt→∞ Lcbb (t), and similarly for the other variables. Later, we shall choose an that produces a contradiction. Since the inequality in (3.18) is strict, Lcbb(t) < for all t sufficiently large. Since the integrand of the integral term in (2.12) increases with Lalb (ξ), from that equation we obtain Z t−τcbb B dAcbb (t) a ≤ −µcbbAcbb (t) + exp(−µlcbb(t − ξ)) dξ dt hτcbb t−2τcbb B ≤ −µacbbAcbb (t) + h and therefore Acbb ≤

B . hµacbb

Also for t sufficiently large, using Lcbb (t) < we find from (2.7) that dAalb (t) ≥ −µaalb Aalb (t) + exp(−(µlalb + eσ)τalb )b(Aalb (t − τalb )). dt By assumption (3.1), if is sufficiently small then there exists A∗∗ alb () > 0 such that exp(−(µlalb + eσ)τalb )b(A) > µaalb A

when

0 < A < A∗∗ alb ();

exp(−(µlalb + eσ)τalb )b(A) < µaalb A

when

A > A∗∗ alb ().

∗∗ Moreover, A∗∗ alb () → Aalb as → 0. Since b(·) is increasing, we may apply a comparison argument similar to that described in the proof of Theorem 3.1 to conclude that

Aalb := lim inf Aalb (t) ≥ A∗∗ alb (). t→∞

(3.19) ?120413_10?

13


For t sufficiently large, we use Lcbb (t) < in (2.6). Then, taking the limit inferior as t → ∞, using (3.19) and that b(·) is increasing, we obtain Lalb ≥

b(A∗∗ alb ()) := L∗∗ 1 − exp −(µlalb + eσ)τalb alb () l µalb + eσ

(3.20) ?120413_11?

where the underbar is defined by Lalb = lim inf t→∞ Lalb (t). Note that L∗∗ alb () → ∗∗ L∗∗ as → 0. From (3.20), L (t) ≥ L () − for t sufficiently large. Using that alb alb alb the integrand in (2.11) increases with Lalb (ξ) we obtain from that inequality Z t Z η B eσ(L∗∗ alb () − )Lcbb (ξ) Lcbb (t) ≥ dξ dη. exp(−µlcbb (t − ξ)) τcbb t−τcbb η−τcbb 1 + heσ(L∗∗ alb () − ) (3.21) ?120413_12? We now use another comparison argument to show that Lcbb (t) grows exponentially with t, which contradicts (3.18). We achieve this by a study of the characteristic equation of the integral equation associated with (3.21), i.e. equation (3.21) with = in place of ≥. That characteristic equation is 2 Beσ(L∗∗ l l 2 alb () − ) . (3.22) ?120413_13? 1 − exp −(λ + µcbb )τcbb (λ + µcbb ) = τcbb (1 + heσ(L∗∗ alb () − )) The time has come to choose . We choose so small that 2 l Beσ(L∗∗ l 2 alb () − ) (µcbb ) < 1 − exp −µcbb τcbb τcbb (1 + heσ(L∗∗ alb () − ))

(3.23) ?120413_14?

holds, which is possible because of (3.17). Since (3.23) holds, the left hand side of (3.22) is less than the right hand side when λ = 0. Since the left hand side grows unboundedly with λ while the right hand side tends to a constant, it follows that (3.22) has a positive real root. Therefore, Lcbb(t) grows exponentially, contradicting (3.18). ∗∗ The above claim shows that both M1 = (0, 0, 0, 0) and M2 = (L∗∗ alb , Aalb , 0, 0) are uniform weak repellers for M0 in the sense that lim sup kΦ(t)φ − Mi k ≥ for all φ ∈ M0 , i = 1, 2, t→∞

with the maximum norm k · k. Define a continuous function p : M → R+ by p(φ) = min(φ1 (0), φ2 (0), φ3 (0), φ4 (0)),

∀φ = (φ1 , φ2 , φ3 , φ4 ) ∈ M.

Thus, p is a generalized distance function for the semiflow Φ(t) (see Definition 1.3.1 [32]). It then follows from Theorem 1.3.2 of [32] that there exists an η > 0 such that min{p(ψ) : ψ ∈ ω(φ)} > η for any φ(θ) 6≡ 0 on [−τalb , 0]. Hence, lim inf Lalb (t) ≥ η, lim inf Aalb (t) ≥ η, lim inf Lcbb(t) ≥ η and lim inf Acbb (t) ≥ η t→∞

t→∞

t→∞

t→∞

uniformly for all solutions with Aalb (θ) 6≡ 0 on [−τalb , 0]. 3.5. Spatial spread. Our model does not explicitly include spatial dependence, but we may make some inferences about the spatial spread rate of the infestation when the system is in the steady state with the CBB present, and a useful formula may be obtained for the spread rate in terms of measurable parameters. We feel it is inappropriate to try to use simple standard approaches to dispersal modelling, for example, by considering a reaction-diffusion extension of the model. One reason for this is that it seems unlikely the ALB would disperse in a random walk manner since beetles tend to reattack the same host tree year after year, laying eggs further down in successive years, and moving to a new host only when

14


a tree has become too heavily infested. In that case the motion will be directed towards previously uninfested forest regions. In ALB infestations in New Jersey the infestations remained localised with very slow spread until the local resource had become over-exploited, which caused beetles to fly much greater distances, of up to 1-2km, in search of new hosts ([9], and references therein). Mark-release-recapture studies have demonstrated that beetle dispersal from release trees is positively associated with the abundance of beetles at the release tree (Bancroft and Smith [2]), a phenomenon that simple diffusion models cannot capture. Another concern with simple reaction-diffusion models is the problem of measuring the diffusion coefficients associated with the dispersal terms that arise from random walk arguments because to do so would involve the release of the pest, which is prohibited in some countries including the USA. Mark and release studies in Gansu Province, China suggest that dispersal of the ALB depends on the spacing of suitable host trees (Smith et al [24]). The beetles will fly further if few host trees are present in the surrounding area, than if host tree density is high. We account for this through our tree density parameter T in the analysis below. Yet another relevant factor is that the ALB continually destroys its habitat since infested trees die, and the adult beetles must therefore fly in search of new host trees. Continual destruction of the habitat, making previously infested regions uninhabitable to future beetle generations, is a feature not taken account of in reaction diffusion extensions of standard population models such as Fisher’s equation. Let us assume that the beetle habitat consists of a thin, effectively one dimensional, semi-infinite forest Ω = {(x, y) : 0 ≤ x < ∞, 0 ≤ y ≤ w}. Beetles are introduced at the end x = 0, the forest width is w, the distribution is uniform in the y-direction and beetles do not leave the forest. After a while, we assume that adult ALB beetle numbers have settled down to the steady state value A∗alb dealt with in subsection 3.2.2. Although each year some beetles move in the positive x-direction in search of new host trees, this phenomenon is consequent upon the loss of previously infested trees that are no longer potential hosts. We may imagine that, perhaps after some initial transient, there is an interval of x-values, starting at x = 0, containing only dead or abandoned trees. Adjoining that interval is another finite interval of fixed length containing infested trees with active ALB activity, and beyond that interval is the rest of the forest containing a certain density of potential host trees, and possibly other trees as well. The ALB population is at steady state, occupying an interval of x values of fixed length, and that interval moves along the x axis with a certain constant speed that we calculate. In this simple way we study the spatial spread even though the model has no explicit spatial dependence. Let T denote the tree density, the mean number of suitable host trees per unit area. Let N denote the mean number of hatched eggs that can be accommodated in a particular tree before that tree has extensive damage and is no longer a suitable host. Let x = X(t) be the location of the boundary of the spreading infestation, i.e. the point beyond which there are no infested trees. We assume for the sake of simplicity that when new host trees are being sought, adult ALB will lay on a particular new host the maximum number N of eggs that the tree can accommodate beyond which it is a doomed host. The way the ALB use hosts could be more wasteful, involving the laying of fewer eggs on a host than could be accommodated before looking elsewhere. This would result in a more rapid invasion of the forest. However, we do know (Smith et al [24]) that adult ALB prefer to remain on their host tree, moving only if necessary. So we assume optimal use of hosts, with the understanding that what we are calculating is a lower bound on the invasion speed. With these assumptions, at any instant in time, an interval of x values consists of infested trees each of which is maximally infested and is not a potential future host. As and when ALB mature from those trees, they must move in the positive x-direction in search of new hosts. The steady state egg laying rate is b(A∗alb ), which


15

we redefine as the egg laying rate per unit width. In a small interval of time [t, t+δt] the newly matured adults move to find new hosts, redefining the boundary of the infestation as x = X(t + δt), and lay wb(A∗alb ) δt eggs in a previously uninfested region of area (X(t + δt) − X(t))w. That region contains T (X(t + δt) − X(t))w new host trees and can accommodate N T (X(t + δt) − X(t))w eggs. Thus N T (X(t + δt) − X(t))w = wb(A∗alb ) δt and, taking the limit as δt → 0, we find the invasion speed to be dX(t) b(A∗alb ) = . dt NT

(3.24) ?170413_1?

Here we assume that ALB have essentially unlimited food and space as an invasive species, and will therefore find hosts for as many eggs as they want to lay. In two spatial dimensions one imagines the forest region with active ALB-infestation to be an annular region of growing radius enclosing a dead forest. Here, if b(A∗alb ) is redefined as the egg laying rate per unit circumference, similar modelling leads again to the right hand side of (3.24), this time interpreted as the rate of change of the outer radius of the annular region of active infestation. Further insight can be gained from the use of the approximate expressions derived in subsection 3.2.2 for the equilibrium values when B is large. In this case, b(A∗alb ) ≈ b0 (0)A∗alb . Using this approximation, and also (3.9) and (3.4), expression (3.24) for the invasion speed can be approximated for large values of B by ln(b0 (0)/µaalb )τcbb (µlcbb )2 . 2 τalb (1 − µaalb /b0 (0)) Beσ(1 − exp(−µlcbb τcbb ))2 − heστcbb (µlcbb ) N T

Thus, the invasion speed decreases when host tree density T increases.

4. Simulations and discussion. Suppose the egg laying rate for the ALB is bA chosen as b(A) = 1+aA , where the regulation parameter a is dependent on the tree type. Female ALB typically mate and lay eggs throughout their lifetime. Individual females appear to be capable of laying between 30 and 127 eggs [3], with a monthly estimate of between 14 eggs per female to 90. This wide variation in estimates is partially due to the fact that fecundity is strongly dependent on the host tree type. Norway maple attracts large numbers of eggs, while egg laying rates for black willow are at the lower end of the scale [9]. On the assumption that 50% of adults are females, we chose to use a monthly fecundity rate of b = 28. The value of 1/µaalb represents the mean adult ALB longevity, and [3, 9] report estimates of between 3 and 137 days. Taking three months as the longevity estimate, we obtain µaalb = 1/3 per month. In China A. glabripennis requires 1 − 2 years to develop from egg to adult and generally overwinters as a larva, depending on the type of host tree in which the larva develops [9]. In the region of Inner Mongolia, in the north of the country, a generation takes two years to develop whereas just one year has been reported for Taiwan ([9], and references therein) suggesting a correlation with latitude. It is estimated that overall in China about 80% of individuals can complete their development within 1 year and fewer than 20% require 2 years [9]. Therefore, we use the estimate τalb = 12 months. )(1− The survival probability of larvae was reported in [31] to be (1− 101+74+139 1282 14+14 ). Factors harmful to larval survival include low temperature, fungi, bacteria, 682 mechanical damage and attack by woodpeckers. Therefore, the natural per-capita mortality rate µlalb of the ALB larvae can be computed from survival probability = exp(−µlalb τalb ).

16


From this, we estimate that µlalb = 0.0269 per month. The mean duration of the CBB Dastarcus helophoroides larval stage is 8.4 days [11], which will parasite in the host ALB. Therefore, τcbb = 8.4 days. The survival probability for the larvae is estimated to be 60.7% [11], based on which we can compute the per-capita CBB larval death rate µlcbb , in units of day−1 , from 60.7% = exp(−µlcbb × 8.4). After converting to a monthly rate, we arrive at a per-capita mortality rate for larval CBB of µlcbb = 1.783 per month. In general, one late-instar larva of ALB can support a complete development of 10 − 35 D. helophoroides larvae [29]. Therefore, we assume B = 10. Since e, the encounter rate, depends on the movement velocity of the consumer and its radius of detection of food items, we suppose it takes 0.5 days to locate food, that is, e = 0.5/30. The parameter σ, representing the fraction of food items encountered that a CBB ingests, is taken as 1 here. The handling time h, which incorporates the time required for the digestive tract to handle the item, is supposed to be equal to the maturation delay of the CBB, that is, h = 8.4/30 months. 4

3

x 10

6000

2.5

5000

1.5

4000

3000

1

2000

0.5

1000

0

?hFig:WithoutCBBsi?

number of A

number of L

alb

alb

2

0

50

100

150 time

200

250

300

0

0

50

100

150 time

200

250

300

Fig. 4.1. Evolution of the numbers of larval and adult ALB in the case when the CBB are absent.

Fig. 4.1 shows that, in the absence of the CBB control agent, the number of ALB will stabilize at a high constant level. With the introduction of CBB, the number of ALB is significantly reduced and the numbers of ALB and CBB oscillate due to predation of CBB on ALB (Fig. 4.2). The growth of the infestation zone is illustrated, in the cases of CBB absent and CBB present, in Figs. 4.3 and 4.4 respectively. In these simulations we set N = 100 and T = 100 in equation (3.24). The prediction is that the infested region grows rapidly in the absence of CBB introduction, its boundary advancing by 35 km in 10 years. However, if the control agent CBB is introduced then (3.24) predicts that the boundary advances less than 1 km over the same time period. The conclusions that can be drawn from the mathematical analysis are as follows and would apply either to the CBB or another candidate control agent comparable to the CBB. The equilibrium analysis of Section 3.2.2, which deals with the situation when the control agent is present, is rather implicit but we do have the useful explicit expression (3.4) for the steady state number of larval ALB when under control by CBB. That expression shows that either the CBB or another candidate control agent should have a conversion efficiency B that is as high as possible. However, obviously the CBB itself has to persist and not be driven to extinction. The conditions for CBB-persistence are the same as those for existence of an equilibrium in which the CBB and ALB coexist. Assumption (3.1) ensures that the ALB would persist in the absence of the control agent, so let us focus on how to interpret (3.7).

17


450

100

400

90 80

350

70 number of Aalb

number of Lalb

300 250 200 150

50 40 30

100

20

50 0

60

10 0

50

100

150 time

200

250

300

0

50

100

150 time

200

250

300

0

0

50

100

150 time

200

250

300

4500 4000 3500

number of L

cbb

3000 2500 2000 1500 1000 500 0

Fig. 4.2. Simulation, in the case when CBB are present, of the numbers of larval and adult ALB, and larval CBB. ?hFig:WithCBBsi?

Fig. 4.3. Spread of ALB infestation in the case when the control agent CBB is absent. ?hFig:Spread_withoutCBBi?

Fig. 4.4. Spread of ALB infestation when CBB are present. ?hFig:Spread_withCBBi?

18


Note that (3.7) is violated if τcbb is large, and also if τcbb is small since the left hand side of (3.7) is O(τcbb ) as τcbb → 0. Thus the CBB can only survive if their larval developmental time is neither too large nor too small. If it were too large not enough CBB would mature. If it is too small the interpretation is that they fail to consume enough ALB biomass. More care on interpretation is needed here since pupation might occur on reaching a critical mass, as is the case for the ALB itself. This consideration suggests some kind of connection between the maturation time τcbb and the digestion (handling) time per unit biomass consumed, h. However, the relationship is unlikely to be a simple one because total maturation time includes time spent searching for food which, in turn, depends on availability of ALB larvae. An alternative type of model formulation involving threshold type delay equations could be appropriate here. We may note that the left hand side of (3.7) is made bigger by decreasing h and this increases the chance of (3.7) being satisfied. Moreover, expression (3.4) indicates that a low value of h makes sense from the point of view of minimizing numbers of ALB larvae. Thus, we want a candidate control agent such as the CBB to have a high conversion efficiency B, a low handling time h and a maturation time τcbb that is neither too high nor too low, in the sense that it satisfies (3.7). The point is that a combination of a low handling time with an intermediate maturation delay allows for the consumption of considerable amounts of larval ALB biomass, which is then converted into CBB biomass. Finally, we note that if the per-capita larval CBB mortality parameter µlcbb is very small then, on approximating the exponential using exp(−x) ≈ 1 − x, inequality (3.7) can be rewritten as τcbb >

1 + heσL∗∗ alb BeσL∗∗ alb

(4.1) ?230913_1?

with L∗∗ alb defined in (3.3), and then (3.4), by which we estimate the ALB infestation as measured by L∗alb when the control agent CBB is present, becomes L∗alb ≈

1 Beστcbb − heσ

provided Bτcbb > h, which holds automatically if (4.1) holds. Acknowledgments. The authors would like to thank Dr. Daihai He for helpful discussions. We are also grateful to two anonymous referees for their careful reading and helpful suggestions. REFERENCES AllenVictory [1] E.J. Allen and H.D. Victory, Modelling and simulation of a schistosomiasis infection with biological control, Acta Trop., 87(2003), 251-267. Bancroft05 [2] J.S. Bancroft and M.T. Smith, Dispersal and influences on movement for Anoplophora glabripennis calculated from individual mark-recapture, Entomologia Experimentalis et Applicata, 116(2005), 83-92. RiskAnal [3] S.M. Bartell and S.K. Nair, Establishment risks for invasive species, Risk Anal., 24(2003), 833-845. CourchampSugihara [4] F. Courchamp and G. Sugihara, Modeling the biological control of an alien predator to protect island species from extinction, Ecol. Appl., 9(1999), 112-123. FaganLewisNeubertPauline [5] W.F. Fagan, M.A. Lewis, M.G. Neubert and P. van den Driessche, Invasion theory and biological control, Ecol. Lett., 5(2002), 148-157. StephenZouSIAP [6] S.A. Gourley and X. Zou, A mathematical model for the control and eradication of a wood boring beetle infestation, SIAM J. Appl. Math., 68(2008), 1665-1687. ManageRevEntomol [7] R.A. Haack, F. H´ erard, J. Sun and J.J. Turgeon, Managing invasive populations of Asian longhorned beetle and Citrus longhorned beetle: A worldwide perspective, Annu. Rev. Entomol., 55(2010), 521-546. HsuHwangKuang [8] S.-B. Hsu, T.-W. Hwang and Y. Kuang, A ratio-dependent food chain model and its applications to biological control, Math. Biosci., 181(2003), 55-83.


19

EcoChinaRev [9] J. Hu, S. Angeli, S. Schuetz, Y. Luo and A.E. Hajek, Ecology and management of exotic and endemic Asian longhorned beetle Anoplophora glabripennis, Agric. For. Entomol., 11(2009), 359-375. Kuang93 [10] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, Academic Press, 1993. BioFeature [11] Q. Lei, M. Li and Z. Yang, A study on biological feature of Dastarcus longulus, J. Northwest Sci-Tech Univ. Agric. For., 31(2003), 62-66. MarkPauline [12] M.A. Lewis and P. van den Driessche, Waves of extinction from sterile insect release, Math. Biosci., 116(1993), 221-247. Biocontrol [13] M. Li, Y. Li, Q. Lei and Z. Yang, Biocontrol of Asian longhorned beetle larva by releasing eggs of Dastarcus helophoroides (Coleoptera: Bothrideridae), Sci. Silvae Sin., 45(2009), 78-82. LouZhaoBMB [14] Y. Lou and X.Q. Zhao, Modelling malaria control by introduction of larvivorous fish, Bull. Math. Biol., 73(2011), 2384-2407. MartinPNAS [15] E.A. Martin, B. Reineking, B. Seo and I. Steffan-Dewenter, Natural enemy interactions constrain pest control in complex agricultural landscapes, PNAS, 110(2013), 5534-5539. MillsGetz [16] N.J. Mills and W.M. Getz, Modelling the biological control of insect pests: a review of host-parasitoid models, Ecol. Model., 92(1996), 121-143. Mischaikow95 [17] K. Mischaikow, H. Smith and H.R. Thieme, Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions. Trans. Amer. Math. Soc. 347(1995), 1669-1685. OliveiraHilker [18] N.M. Oliveira and F.M. Hilker, Modelling disease introduction as biological control of invasive predators to preserve endangered prey, Bull. Math. Biol., 72(2010), 444-468. Smith95 [19] H.L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, AMS, Providence, RI, 1995. SmithMT00 [20] M.T. Smith, Biocontrol and IPM for the Asian longhorned beetle, The IPM Practitioner, 22(2000), 1-5. SmithMTfeature [21] M.T. Smith, The Potential For Biological Control of Asian Longhorned Beetle in the U.S. http://www.entomology.wisc.edu/mbcn/fea606.html (accessed Sept 21, 2013). SmithMT01 [22] M.T. Smith and Z.Q. Yang, Investigations of natural enemies for biocontrol of Anoplophora glabripennis (Motsch.), U.S. Dep. Agric. For. Serv. Gen. Tech. Rep., NE-285: 139–141, 2001. SmithMT03 [23] M.T. Smith, Z.Q. Yang, F. H´ erard, R. Fuester, L. Bauer, L. Solter, M. Keena and V. D’Amico, Biological control of Anoplophora glabripennis (Motsch.): A synthesis of current research programs, U.S. Dep. Agric. For. Serv. Gen. Tech. Rep., NE-285: 87–91, 2003. Smith04 [24] M.T. Smith, P.C. Tobin, J. Bancroft, G. Li and R. Gao, Dispersal and spatiotemporal dynamics of Asian Longhorned Beetle (Coleoptera: Cerambycidae) in China, Environ. Entom., 33(2004), 435-442. TangCheke [25] S. Tang and R.A. Cheke, Models for integrated pest control and their biological implications, Math. Biosci., 215(2008), 115-125. ThiemeJMB [26] H.R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30(1992), 755-763. Thieme03 [27] H.R. Thieme, Mathematics in population biology, Princeton University Press, Princeton, NJ, 2003. UMDBull [28] UMD Entomology Bulletin, Asian Longhorned Beetle, Exotic Pest Threats, 2008. Limone [29] J.-R. Wei, Z.-Q. Yang, H.-L. Hao and J.-W. Du, (R)-(+)-limonene, kairomone for Dastarcus helophoroides, a natural enemy of longhorned beetles, Agric. For. Entomol., 10(2008), 323-330. Yang04 [30] Z.Q. Yang, M. Li and M.T. Smith, The bionomics and population dynamics of Dastarcus longulus (Coleoptera: Colydiidae), In: Proceedings of the 15th International Plant Protection Congress, Beijing, 2004. Lifetable [31] R. Zhou, Z. Lu, X. Lu and X. Wu, Life table study of Anoplophora glabripennis (Coleoptera: Cerambycidae) natural populations, J. Beijing Forestry Univ., 15(1993), 125-129. ZhaoBook [32] X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, 2003.