Modelling Integrated Pest Management Strategies

Robert A. ChekeNRI, University of Greenwich at Medway(r.a.cheke@greenwich.ac.uk)

Department of Mathematical Sciences, University of Greenwich13 March 2019

Professors Yanni Xiao1 & Sanyi Tang2

1Department of Applied MathematicsXi’an Jiaotong University, Xi’an 710049People’s Republic of China

2School of Mathematics and Information ScienceShaanxi Normal UniversityXi’an 710119People’s Republic of China

The Pesticide Treadmill

• Farmers in the USA lost 7% of their crops to pests in the 1940s, but since the 1980s the percentage lost has increased to 13%, even though more pesticides were being used

• > 500 species of pests have now developedresistance to pesticides

• Often caused by the same classes of pesticides being used repeatedly for a long time.

• Other problems ensue such as pest resurgence, acute and chronic health problems, environmental pollution and uneconomic crop production.

CLASSICAL BIOLOGICAL CONTROLNATURALUses:• Pathogens (Micro-organisms which cause disease in insects)• Parasites (An organism which lives on or in a host, including both

microbes and multicellular organisms)• Parasitoids (Insect parasites with immature stages that develop on or

within a host, which is destroyed as the parasitic larva completes its development)(a) Primary Parasitoids

Develop in or on non-parasitic hosts(b) Hyperparasites

Develop on or in another parasitoid• Predators

OTHER FORMS OF BIOLOGICAL CONTROL• HOST PLANT RESISTANCE• CULTURAL CONTROL• STERILE-INSECT CONTROL METHOD• INTRODUCTION OF COMPETITORS TO PESTS• GENETIC CONTROL OF PESTS• MANIPULATION OF ENDOSYMBIONTS IN PEST

POPULATIONS

Integrated Pest Management

• Integrated pest management (IPM) is a long-term control strategy that combines biological, cultural, and chemical tactics to reduce pest populations to tolerable levels when the pests reach an economic threshold (ET).

• IPM has been shown experimentally to be more effective than classical methods such as biological or chemical control only.

Integrated Pest Management

Thresholds• Economic injury level (EIL) =

lowest population density that will cause economic damage

• Economic threshold (ET) = population density at which control measures should be initiated to prevent an increasing pest population from reaching the economic injury level

The arrows indicate points where pest levels exceed the economic threshold (ET) and an IPM strategy should be applied.

Some IPM strategies for Alfalfa pests

IPM models based on Biological Control Models

• Population models of hosts and parasitoids(Thompson, Nicholson & Bailey models)

• Population models of predators and prey(Lotka-Volterra predator-prey models)

• Population models of competition(Lotka-Volterra competion models)

The Nicholson & Bailey model

SIMULATIONS OF CO-EXISTING SYMPATRIC SPECIES AND STABILISING EFFECTS OF HOST SPATIAL HETEROGENEITY AND MUTUAL INTERFERENCE BETWEEN PARASITOIDSModel outputs showing coexistence under different conditions of Mesopsocus unipunctatus and M. immunis parasitised by Alaptusfusculus and Leiophron sp. on larch trees in Yorkshire

BROADHEAD, E. & CHEKE, R. A. (1975) Host spatial pattern, parasitoid interference and the modelling of the dynamics of Alaptus fusculus (Hym.:Mymaridae), a parasitoid of two Mesopsocus species (Psocoptera). J.Anim.Ecol. 44: 767-793.

Includes density dependent effect on parasitoid spatial distributions

Mesopsocus unipunctatusPhoto:http://www.brc.ac.uk/schemes/barkfly/images/photos/Mesopsocus%20unipunctatus_BS.jpg

Mesopsocus immunisPhoto:http://www.ispotnature.org/node/171410

Questions investigated by analytical modelling and numerical simulations

• Effects of delayed responses to pesticides?• Optimal timing of interventions?• Optimal numbers of biocontrol agents to

introduce?• Effects of pesticide resistance• Etc etc.

What are the characteristics of a good biological control agent?

• Why are they usually parasitoids not predators?• Polyphagy?• Threshold of prey consumption, below which they

cannot reproduce, so no equilibrium possible?• Is it better to introduce one, two or many agents?• Why do ideal biological control agents have a high

search rate, an ability to aggregate in patches of high host density and short handling times relative to their longevity?

A difference equation host-parasitoidmodel (model 2)

• where Ht and Pt are the host and parasitoid population abundance in generation t

• b2(H) is the per capita net rate of increase of the host population

• g(H,P) is the proportion of host individuals that escape attack by the parasitoid

• γ2 embodies the numerical response or the average number of parasitoids that emerge per host individual parasitized

• δ (δ > = 0) denotes the density-independent survival of parasitoidpropagules at generation t.

• The parasitoid intergenerational survival rate can be affected by several different factors such as immigration from outside the local area, parasitoid overwintering survival and parasitism of an alternate insect pest which is not modelled explicitly.

Functional response• The functional response of a consumer to a change in the density of a resource is the rate at which

an individual consumer extracts resources as a function of resource density.• Holling presented three different functional response classes:

type I (linear then constant)type II (decelerating rise to an upper asymptote)type III (sigmoidal).

• The most widely used model for describing the functional response of parasitoids is the disc equation of Holling’s type II.

• Assuming that the production of hosts (in the absence of parasitoids) follows the usual logistic growth, i.e.

where r is the intrinsic growth rate, and K is the carrying capacity of the environment.• Further assume that the host-dependent functional response is of the Holling’s type II form, i.e.

where α is the instantaneous search rate, i.e. the average number of encounters with hosts per parasitoid per unit of searching time, T the total time initially available for search, i.e. the total time that the hosts are exposed to parasitoids and Th the handling time, i.e. the time between a host being encountered and search being resumed.

Extension of H-P model to include IPM

Host–parasitoid populations in model (7) periodically oscillate with period 5, and the sequence of Tnt (times

when the solution reaches ET) shown at the bottom implies that the instant killing rate of the host population by an insecticide application stabilizes at 0.221, which provides the dosages and timing of insecticide applications.

From the biological point of view, the effective testing of parasitoid release strategies and timing of insecticide applications require the use of replicated treatments in greenhouses and the use of experimental controls either in cages or separate greenhouses. Thus, experimental methods in combination with the model approaches are the most cost effective method.

Different host intrinsic growth rates and different parasitoid characteristics may have the same mean outbreak period. But a large r results in a high instant killing rate(Fig. 5(A) and (B)), while efficient searchers (large α) reduce the instant killing rate (Fig. 5(C) and (D)).

Effects of different initial conditions (with ET = 35)

The trajectory shown in Fig. 7(A) with initial densities (H0,P0) = (10,1) indicates that the solution approaches a steady state after one integrated control application.With initial densities as (10, 7), the trajectory shown in Fig. 7(B) shows that the host population will have outbreaks an infinite number of times and a periodic control strategy is needed. But if we increase the initial density of the parasitoid population to 20, the solution is free of impulsive effect and directly tends to the steady state (Fig. 7C). In this case, the natural enemies can successfully suppress the host outbreak and result in a stable system between host and parasitoid. If the initial density of the parasitoid is further increased to 40, Fig. 7(D) implies that a periodic control strategy is needed to control the host population and avoid economic damage. These results confirm that inundative release of a parasitoidpopulation is not necessarily beneficial for insect pest control.

Effects of different host intrinsic growth rates

To investigate how the parasitoidintergenerational rate δ affects control programmes, we tested several different δvalues, and the simulation results imply that model (7) has strong periodicity even with high intrinsic growth rate, r, if δ is relative small.

However, the dynamical behaviour becomes more and more complex as δincreases. It follows from Fig. 9 that if the parasitoid intergenerational survival rate is large enough (here δ = 0.8), the model (7) has complex dynamical behaviour even with a very low host intrinsic growth rate. In particular, the time series shown at the bottom of Fig. 9 implies that the host outbreak period and instant killing rate are extremely complex. From a biological point of view, a high parasitoidintergenerational survival rate means a high initial parasitoid density in the next generation, and then the high initial density of parasitoids may counter-intuitively lead to more severe host outbreaks.

Above from TANG, S. & CHEKE, R.A. (2008) Models for integrated pest control and their biological implications. Mathematical Biosciences 215: 115-125. (doi:10.1016/j.mbs.2008.06.008).

Paper includes analogous results for a continuous version of the model.

Predator-Prey Relationships: theory• Lotka-Volterra Predator-Prey models• Assume N prey and P predators• Assume prey population changes according to

dN/dt = αN – βNP

where α is the prey’s intrinsic rate of increase in the absence of the predator and β is the capture rate of prey per predator

• Assume prey population changes according to

dP/dt = δNP – γP

where δ is the rate at which predator converts prey into predator births and γ is the rate of predator deaths

Lotka-Volterra Predator-Prey models• So combining the two population models we get two differential

equations:

dN/dt = αN – βNPdP/dt = δNP - γP

• Assumptions:

(a) The prey population finds ample food at all times

(b) The food supply of the predator population depends entirely on the size of the prey population

(c) The rate of change of a population is proportional to its size

(d) During the process, the environment does not change in favour of one species and genetic adaptation is inconsequential

(e) Predators have limitless appetite

Lotka-Volterra Predator-Prey models

Leads to limit cycle oscillations with prey populations leading the predator populations

IPM model with Predator-prey function

Similarity with Lotka-Volterra system

IPM Impusive differential equation

Proof involves the comparison theorem on impulsive differential equations (Lakshmikantham et al. 1989)

ResultsA typical solution of the predator-pest system with impulsive effects is shown in Fig. 2. The variable y(t) oscillates in a stable cycle [Fig. 2(a)]. In contrast, the pest x(t) rapidly decreases to zero [Fig. 2(b)] and Tmax = 3.556.

If the period of pulses T is larger than Tmax, the pest-eradication solutionbecomes unstable and the variable x(t) begins to oscillate with a large amplitude,corresponding to periodic bursts of the pest population. If the period of pulses is further increased, a sequence of ‘period adding’ bifurcations interchanging with regions of chaos is observed. A typical chaotic attractor is shown in Fig. 3.

TANG, S., XIAO, Y., CHEN, L. & CHEKE, R. A. (2005) Integrated pest management models and their dynamical behaviour. Bulletin of Mathematical Biology 67: 115-135. (doi:10.1016/j.bulm.2004.06.005).

Main mathematical result of Tang et al. (2005)

• “Using analytical methods, we show that there exists an orbitally asymptotically stable periodic solution with a maximum value no larger than the given Economic Threshold (ET). The complete expression for this periodic solution is given and the ET is evaluated for given parameters.

• We also show that in some cases control costs can be reduced by replacing IPM interventions at unfixed times with periodic interventions. Further, we show that small perturbations of the system do not affect the existence and stability of the periodic solution. Thus, we provide the first demonstration using mathematical models that an IPM strategy is more effective than classical control methods.”

Effects of pesticide resistance• Resistance occurs pest is exposed to a

pesticide and not all of the insects are killed.

• Those individuals that survive are often genetically predisposed to be resistant to the pesticide.

• Repeated applications and higher doses will kill increasing numbers, but some resistant insects will survive.

• The offspring of the survivors, carrying the genetic make-up of their parents, will inherit the ability to survive the exposure to the insecticide and will increase in proportion with each succeeding generation. Therefore the total number of insect pests x in system (3.1) must be divided into two classes, susceptible insects and resistant insects, denoted by x1 and x2 respectively. Then we have:

where m is the resistance parameter which depends on the dose of the pesticide andthe impulsive period T. We choose parameter values such that x tends to zero if m = 0. In order to illustrate the effects of pest resistance on the pest control, we can, for instance, set m = 0.05 and the initial conditions satisfy x1(0) > 0, x2(0) = 0 and y(0) > 0.

The pest population oscillates with a large amplitude, corresponding to its periodic bursts. Hence need to monitor pest populations for resistance and incorporate resistance management into integrated control measures.

In practice, amounts of pesticides used are increased to regain the initial levels of control after pest resistance has emerged. However, the levels of control then seem to decline even more rapidly. Further numerical simulations on the system show this to be the case in theory too.

Prey-dependent consumption model with unfixed moments of an impulsive effect: IPM control interventions (including augmenting natural enemies, spraying pesticides, catching or harvesting the pest) are applied when the pest population reaches an ET

Periodic solution (Proof involving the Lambert W-function omitted.

The Lambert W-function is defined to be a multivalued inverse of thefunction z → zez satisfying W(z) exp(W(z)) = z.)

Inclusion of costs• Results show that an IPM programme is more

cost-effective than either chemical or biological control used on their own.

• However, when the two are used in conjunction as part of an IPM strategy the chemicals must be applied in such a way that they do not kill the beneficial natural enemies in addition to the pests.

Above from: TANG, S., XIAO, Y., CHEN, L. & CHEKE, R. A. (2005) Integrated pest management models and their dynamical behaviour. Bulletin of Mathematical Biology67: 115-135. (doi:10.1016/j.bulm.2004.06.005).

Other topicsOther modelling exercises have considered IPM in relation to:• How many natural enemies to release?• Optimal timing of interventions• Phase changes of migrant pests such as locusts• Thresholds for optimal switching times between deployment of

different pesticides• Optimal ratios of Bt-corn to non-Bt- corn to be planted with a mixed

planting strategy • Effects of delayed responses to pesticides• Effects of residual activity of pesticides• Effects of predator and prey dispersal• Effects of prey refuges• Sliding mode control for IPM

Modelling control of mosquito dengue vectors by supplanting them with Wolbachia-infected

populationsRobert A. Cheke

Natural Resources Institute, University of Greenwich at Medway, Central Avenue, Chatham Maritime,

Chatham, Kent, ME4 4TB, UKr.a.cheke@greenwich.ac.uk

Xianghong Zhang & Sanyi TangCollege of Mathematics and Information Science,

Shaanxi Normal University, Xi’an, 710062, P.R. China

Outline

♣ How birth-pulse affects different strategies

♣ Augmentation with mosquitoes withthe same sex ratio as the target population

♣ Biological background

♣ Augmentation with mosquitoes withdifferent sex ratios from the target population

Dengue fever and dengue haemorrhagic fever

Causative agent: dengue virus, a flavivirus that occurs as 5 different serotypes (DENV-5 discovered in Malaysia in samples taken in 2007; see D Normile - Science, 2013) Vectors

• Aedes aegypti

• Aedes albopictus

• Aedes polynesiensis• Aedes scutellaris

Source: http://www3.imperial.ac.uk/newsandeventspggrp/imperialcollege/newssummary/news_30-1-2015-17-9-41

Human antibodies attacking dengue virus

Different diseases also spread by dengue vectors

Aedes aegypti• Yellow Fever• Chikungunya virus• Zika virus

• ? Venezuelan Equine Encephalitis virus

• ? West Nile virus

Aedes albopictus (Asian Tiger Mosquito)• Chikungunya virus• Dirofilariasis• Zika virus• ?Eastern equine encephalitis

virus• ? La Crosse virus• ? Venezuelan equine

encephalitis virus• ? West Nile virus• ? Japanese encephalitis virus• ? Usutu virus

Distribution and harmfulness for dengue fever

Globalization - up to 100 countries (2/5 population)

nearly 380 million people infected each year

mostly occurring in children and adolescents

mortality rate being less than 0.1%, about 5%-10%

frequently infected and multi-symptom

Except for general dengue fever, there are more severe dengue hemorrhagic fever and dengue shock syndrome.

urbanization, population growth, deterioration of public infrastructure, international travel

Wolbachia(endosymbiosis)

Maternal transmission

Cytoplasmic incompatibility(CI)

Transmission from mother to offspring: one type of vertical transmission.

Parthenogenesis, feminization,killing of males etc.

WolbachiaUp to 65% of insect species naturally carry Wolbachia, which is also found in ticks, mites, spiders and nematodes. About 28% of mosquito species carry various strains of Wolbachia.Strains vary in their characteristics

)10( 6≈

W-Mel strain of Wolbachia in a mosquito ovary

Source: http://www.eliminatedengue.com/library/image/lg/MEL-OVARYB-10X-v2.jpg

Embryo death when uninfected female mates with infected male.

XX

X

X

CI

Wolbachia- infectedWolbachia-uninfected

Maternal transmission

Cytoplasmic Incompatibility (CI):

From: http://www.eliminatedengue.com/our-research/wolbachia

Control strategy

Combinations of CI with maternal transmission, different types of Wolbachia, release methods and ratios of infected female to male mosquitoes may lead to different steady states for mosquitoes, including eradication, coexistence and total replacement.

Population suppression (eradication) : release of infected males (CI)

Population replacement: releasing infected females makes uninfected mosquitoes infected. (CI and maternal inheritance)

Biological control

Wolbachia is not found in vertebrates, so it is unlikely to be transmissable to humans.

Wolbachia can prevent vectors from replicating and transmitting dengue virus to block the spread of dengue diseases.

Wolbachia does not exist in the agents of most insect-borne diseases, e.g. dengue fever and malaria.

control the spread of dengue virus

The first open releases were in Australia. Mosquitoes infected with wMel Wolbachia (strong anti-dengue properties and low fitness costs) were released at Yorkeys Knob and Gordonvale in north-eastern Australia in 2011. success!!!

Open releases of Wolbachia mosquitoes - Australia

Hoffmann et al., 2011, nature.

● In 2015, the worst ever outbreak occurred in China, with more than 47,000 cases, compared with around 1,000 in previous years, especially in Guangdong province (about 42,000 cases). ● Based on the CI mechanism, a science "factory" in Guangdong was opened on July 17th., as the largest mosquito factory in the world, breeding 10 million male mosquitoes infected with Wolbachia every week.● From Mar.12, 2015, Zhiyong Xi, a microbiology professor at Sun Yat-sen University, and his group began to release “sterile mosquitoes” in Shazai island, 3 square km2 in area, in a Guangzhou suburb, three times a week, and every time about 70,000-100,000 mosquitoes were released. In March 2016, releases also on Dadao island (also in Guangzhou).

Open releases in China

The world's largest mosquito factory is based in Guangzhou Science City and producesaround one million of the sterilised insects each week

● The ratio of Wolbachia-infected male mosquitoes to wild males is kept at 5:1, which means that the mating rate of wild females with Wolbachia-infected males is over 80%, and their offspring will not hatch.● The field experiment includes two steps:

The first step aims to reduce the density of mosquitoes by releasing a large number of Wolbachia-infected males. -Population suppression, on going!The second step: release of female mosquitoes infected with Wolbachia, so that they can be established and replace the wild females, which can reduce the transmission of dengue virus. -Population replacement!

Open releases in China

Part I: Birth-pulse models to control dengue fever transmission

How do birth-pulses affect different strategies?

ZHANG, X., TANG, S. & CHEKE, R.A. (2015) Birth-pulse models of Wolbachia-induced cytoplasmic incompatibility in mosquitoes for dengue virus control. Nonlinear Analysis: Real World Applications 22: 236–258. (DOI:10.1016/j.nonrwa.2014.09.004).

Research motivation:

◄ Birth pulse growth pattern (Cauchley1989) is more appropriate for mosquito populations than continuous reproduction.

◄ Two density dependent death rate functions (strong and weak) are considered.

Birth-pulse models with Wolbachia-induced CI and different density dependent death rate functions

Do different density dependent death rates lead to different control strategies (population replacement or suppression) ?If so, how to improve control purposes with different death rates?

Notation• The total population of mosquitoes N(t) is divided into

Wolbachia-infected and uninfected classes, with the size of each class denoted by I(t) and U(t), respectively.

• We assumed that the ratio of Wolbachia-infected males to infected females is the same as that of uninfected males to uninfected females.

• It is also assumed that both infected and uninfected individuals have the same natural birth (b) and death rates (δ).

• However, infected individuals may suffer an additional loss of natural fitness, denoted by D, meaning that although their reproductive capacity is not reduced compared with uninfected mosquitoes, their death rate does increase.

Notation• Wolbachia is primarily transmitted vertically, i.e. it is mostly passed

from infected females to their offspring. • The transmission is usually not perfect, occurring with a probability

τ ∊ (0,1].• The terms τ bI(nT) and (1-τ )bI(nT) represent infected mothers that

do and do not transmit Wolbachia bacteria to their offspring, respectively.

• The effect of the CI mechanism results in zygotic death of potential offspring with a probability q ∊ [0,1] when an infected male is crossed with an uninfected female, and all other crosses are unaffected.

• The term b(1-qI(nT)/N(nT))U(nT) refers to the quantity of mosquitoes left after the effect of CI.

• The infected and uninfected populations are decreased by an amount (δ+D)I(t)N(t) or ((δ+D)I(t)/(K+N(t))) and δU(t)N(t) (or δU(t)/(K+N(t))), respectively, whenever t ≠ nT.

• Birth pulses with the CI mechanism occur at t = nT.

Model with strong density dependent function (F1)

Model with weak density dependent function (F2)

Birth-pulse models

( ) ( ) ( ) ( ),,

( ) ( ) ( ),(2)

( ) (1 ) ( ),.( )( ) ( ) (1 ) ( ) (1 ) ( ),

( )

dI t D I t N tdt t nT

dU t U t N tdt

I nT b I nTt nTI nTU nT U nT bI nT b q U nT

N nT

δ

δ

τ

τ

+

+

= − + ≠ = − − − − = + = = + − + −

( ) ( )( ) ,( )

,( ) ( ) ,

( ) (3)( ) (1 ) ( ),

.( )( ) ( ) (1 ) ( ) (1 ) ( ),( )

dI t I tDdt K N t

t nTdU t U t

dt K N t

I nT b I nTt nTI nTU nT U nT bI nT b q U nT

N nT

δ

δ

τ

τ

+

+

= − + + ≠ = − + − − − = + = = + − + −

Poincaré (Stroboscopic) maps of systemsThe stroboscopic maps which determine the numbers of infected and uninfected populations immediately after each birth pulse at the discrete times nT play key roles in the investigation of the dynamics of the models.

1 1

1 1

(1 ) ( , ),( ) 1

(7)(1 ) (1 ) ( , ).( ) 1 ( ) 1

nn n n

n n

n n nn n n

n n n n n n

b II f I UT I U

bI I UU b bq g I UT I U I U T I U

τδ

τδ δ

+

+

+ = ≡ + + − − − − = + + − ≡ + + + + +

1 2

1 2

(1 ) ( ) ( , ),(10)

(1 ) ( ) (1 ) ( ) ( , ),

nn n n

n n

n n nn n n

n n n n n n

b KII W z f I UI U

bKI I KUU W z b bq W z g I UI U I U I U

τ

τ

+

+

+ = ≡ + − − − − = + + − ≡ + + +

with exp .n n n nI U I U TzK K

δ+ + − =

For strong density dependence

For weak density dependence

Conclusions from birth-pulse model

• A model with a strong density dependent death function showed that population replacement is possible if the initial proportion of infected mosquitoes exceeds a critical value, especially when Wolbachia transmission from mother to offspring is perfect.

• However, with a weak density dependent death function, conclusions about population eradication are less clear-cut as the system's solutions are sensitive to initial values.

• Nevertheless, using numerical methods it can be shown that population eradication may be achieved regardless of the infection ratio, but only when parameters lie in particular regions and the initial density of uninfected mosquitoes is low enough.

• Further research investigated the effects of introducing infected mosquitoes with different sex ratios and at different ratios to the wild population.