Understanding the Evolutionary Ecology of host–pathogen Interactions Provides Insights into the Outcomes of Insect Pest Biocontrol
Identifieur interne : 000B49 ( Pmc/Curation ); précédent : 000B48; suivant : 000B50Understanding the Evolutionary Ecology of host–pathogen Interactions Provides Insights into the Outcomes of Insect Pest Biocontrol
Auteurs : David J. Páez [États-Unis] ; Arietta E. Fleming-Davies [États-Unis]Source :
- Viruses [ 1999-4915 ] ; 2020.
Abstract
The use of viral pathogens to control the population size of pest insects has produced both successful and unsuccessful outcomes. Here, we investigate whether those biocontrol successes and failures can be explained by key ecological and evolutionary processes between hosts and pathogens. Specifically, we examine how heterogeneity in pathogen transmission, ecological and evolutionary tradeoffs, and pathogen diversity affect insect population density and thus successful control. We first review the existing literature and then use numerical simulations of mathematical models to further explore these processes. Our results show that the control of insect densities using viruses depends strongly on the heterogeneity of virus transmission among insects. Overall, increased heterogeneity of transmission reduces the effect of viruses on insect densities and increases the long-term stability of insect populations. Lower equilibrium insect densities occur when transmission is heritable and when there is a tradeoff between mean transmission and insect fecundity compared to when the heterogeneity of transmission arises from non-genetic sources. Thus, the heterogeneity of transmission is a key parameter that regulates the long-term population dynamics of insects and their pathogens. We also show that both heterogeneity of transmission and life-history tradeoffs modulate characteristics of population dynamics such as the frequency and intensity of “boom–bust" population cycles. Furthermore, we show that because of life-history tradeoffs affecting the transmission rate, the use of multiple pathogen strains is more effective than the use of a single strain to control insect densities only when the pathogen strains differ considerably in their transmission characteristics. By quantifying the effects of ecology and evolution on population densities, we are able to offer recommendations to assess the long-term effects of classical biocontrol.
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DOI: 10.3390/v12020141
PubMed: 31991772
PubMed Central: 7077243
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<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Understanding the Evolutionary Ecology of host–pathogen Interactions Provides Insights into the Outcomes of Insect Pest Biocontrol</title><author><name sortKey="Paez, David J" sort="Paez, David J" uniqKey="Paez D" first="David J." last="Páez">David J. Páez</name><affiliation wicri:level="1"><nlm:aff id="af1-viruses-12-00141">School of Aquatic and Fishery Sciences, The University of Washington, 1122 NE Boat St, Box 355020, Seattle, WA 98195, USA</nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea>School of Aquatic and Fishery Sciences, The University of Washington, 1122 NE Boat St, Box 355020, Seattle, WA 98195</wicri:regionArea></affiliation></author><author><name sortKey="Fleming Davies, Arietta E" sort="Fleming Davies, Arietta E" uniqKey="Fleming Davies A" first="Arietta E." last="Fleming-Davies">Arietta E. Fleming-Davies</name><affiliation wicri:level="1"><nlm:aff id="af2-viruses-12-00141">Department of Biology, University of San Diego, 5998 Alcala Park, San Diego, CA 92110, USA</nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea>Department of Biology, University of San Diego, 5998 Alcala Park, San Diego, CA 92110</wicri:regionArea></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">31991772</idno><idno type="pmc">7077243</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7077243</idno><idno type="RBID">PMC:7077243</idno><idno type="doi">10.3390/v12020141</idno><date when="2020">2020</date><idno type="wicri:Area/Pmc/Corpus">000B49</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Corpus" wicri:corpus="PMC">000B49</idno><idno type="wicri:Area/Pmc/Curation">000B49</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Curation">000B49</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Understanding the Evolutionary Ecology of host–pathogen Interactions Provides Insights into the Outcomes of Insect Pest Biocontrol</title><author><name sortKey="Paez, David J" sort="Paez, David J" uniqKey="Paez D" first="David J." last="Páez">David J. Páez</name><affiliation wicri:level="1"><nlm:aff id="af1-viruses-12-00141">School of Aquatic and Fishery Sciences, The University of Washington, 1122 NE Boat St, Box 355020, Seattle, WA 98195, USA</nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea>School of Aquatic and Fishery Sciences, The University of Washington, 1122 NE Boat St, Box 355020, Seattle, WA 98195</wicri:regionArea></affiliation></author><author><name sortKey="Fleming Davies, Arietta E" sort="Fleming Davies, Arietta E" uniqKey="Fleming Davies A" first="Arietta E." last="Fleming-Davies">Arietta E. Fleming-Davies</name><affiliation wicri:level="1"><nlm:aff id="af2-viruses-12-00141">Department of Biology, University of San Diego, 5998 Alcala Park, San Diego, CA 92110, USA</nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea>Department of Biology, University of San Diego, 5998 Alcala Park, San Diego, CA 92110</wicri:regionArea></affiliation></author></analytic><series><title level="j">Viruses</title><idno type="eISSN">1999-4915</idno><imprint><date when="2020">2020</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><p>The use of viral pathogens to control the population size of pest insects has produced both successful and unsuccessful outcomes. Here, we investigate whether those biocontrol successes and failures can be explained by key ecological and evolutionary processes between hosts and pathogens. Specifically, we examine how heterogeneity in pathogen transmission, ecological and evolutionary tradeoffs, and pathogen diversity affect insect population density and thus successful control. We first review the existing literature and then use numerical simulations of mathematical models to further explore these processes. Our results show that the control of insect densities using viruses depends strongly on the heterogeneity of virus transmission among insects. Overall, increased heterogeneity of transmission reduces the effect of viruses on insect densities and increases the long-term stability of insect populations. Lower equilibrium insect densities occur when transmission is heritable and when there is a tradeoff between mean transmission and insect fecundity compared to when the heterogeneity of transmission arises from non-genetic sources. Thus, the heterogeneity of transmission is a key parameter that regulates the long-term population dynamics of insects and their pathogens. We also show that both heterogeneity of transmission and life-history tradeoffs modulate characteristics of population dynamics such as the frequency and intensity of “boom–bust" population cycles. Furthermore, we show that because of life-history tradeoffs affecting the transmission rate, the use of multiple pathogen strains is more effective than the use of a single strain to control insect densities only when the pathogen strains differ considerably in their transmission characteristics. By quantifying the effects of ecology and evolution on population densities, we are able to offer recommendations to assess the long-term effects of classical biocontrol.</p></div></front><back><div1 type="bibliography"><listBibl><biblStruct><analytic><author><name sortKey="Hajek, A E" uniqKey="Hajek A">A.E. 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<front><journal-meta><journal-id journal-id-type="nlm-ta">Viruses</journal-id><journal-id journal-id-type="iso-abbrev">Viruses</journal-id><journal-id journal-id-type="publisher-id">viruses</journal-id><journal-title-group><journal-title>Viruses</journal-title></journal-title-group><issn pub-type="epub">1999-4915</issn><publisher><publisher-name>MDPI</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">31991772</article-id><article-id pub-id-type="pmc">7077243</article-id><article-id pub-id-type="doi">10.3390/v12020141</article-id><article-id pub-id-type="publisher-id">viruses-12-00141</article-id><article-categories><subj-group subj-group-type="heading"><subject>Article</subject></subj-group></article-categories><title-group><article-title>Understanding the Evolutionary Ecology of host–pathogen Interactions Provides Insights into the Outcomes of Insect Pest Biocontrol</article-title></title-group><contrib-group><contrib contrib-type="author"><name><surname>Páez</surname><given-names>David J.</given-names></name><xref ref-type="aff" rid="af1-viruses-12-00141">1</xref><xref rid="c1-viruses-12-00141" ref-type="corresp">*</xref><xref ref-type="author-notes" rid="fn1-viruses-12-00141">†</xref></contrib><contrib contrib-type="author"><contrib-id contrib-id-type="orcid" authenticated="true">https://orcid.org/0000-0003-1609-7014</contrib-id><name><surname>Fleming-Davies</surname><given-names>Arietta E.</given-names></name><xref ref-type="aff" rid="af2-viruses-12-00141">2</xref><xref rid="c1-viruses-12-00141" ref-type="corresp">*</xref><xref ref-type="author-notes" rid="fn1-viruses-12-00141">†</xref></contrib></contrib-group><aff id="af1-viruses-12-00141"><label>1</label>School of Aquatic and Fishery Sciences, The University of Washington, 1122 NE Boat St, Box 355020, Seattle, WA 98195, USA</aff><aff id="af2-viruses-12-00141"><label>2</label>Department of Biology, University of San Diego, 5998 Alcala Park, San Diego, CA 92110, USA</aff><author-notes><corresp id="c1-viruses-12-00141"><label>*</label>Correspondence: <email>dpaezmc@gmail.com</email> (D.J.P.); <email>arietta.flemingdavies@gmail.com</email> (A.E.F.-D.)</corresp><fn id="fn1-viruses-12-00141"><label>†</label><p>These authors contributed equally to this work.</p></fn></author-notes><pub-date pub-type="epub"><day>25</day><month>1</month><year>2020</year></pub-date><pub-date pub-type="collection"><month>2</month><year>2020</year></pub-date><volume>12</volume><issue>2</issue><elocation-id>141</elocation-id><history><date date-type="received"><day>10</day><month>12</month><year>2019</year></date><date date-type="accepted"><day>17</day><month>1</month><year>2020</year></date></history><permissions><copyright-statement>© 2020 by the authors.</copyright-statement><copyright-year>2020</copyright-year><license license-type="open-access"><license-p>Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (<ext-link ext-link-type="uri" xlink:href="http://creativecommons.org/licenses/by/4.0/">http://creativecommons.org/licenses/by/4.0/</ext-link>).</license-p></license></permissions><abstract><p>The use of viral pathogens to control the population size of pest insects has produced both successful and unsuccessful outcomes. Here, we investigate whether those biocontrol successes and failures can be explained by key ecological and evolutionary processes between hosts and pathogens. Specifically, we examine how heterogeneity in pathogen transmission, ecological and evolutionary tradeoffs, and pathogen diversity affect insect population density and thus successful control. We first review the existing literature and then use numerical simulations of mathematical models to further explore these processes. Our results show that the control of insect densities using viruses depends strongly on the heterogeneity of virus transmission among insects. Overall, increased heterogeneity of transmission reduces the effect of viruses on insect densities and increases the long-term stability of insect populations. Lower equilibrium insect densities occur when transmission is heritable and when there is a tradeoff between mean transmission and insect fecundity compared to when the heterogeneity of transmission arises from non-genetic sources. Thus, the heterogeneity of transmission is a key parameter that regulates the long-term population dynamics of insects and their pathogens. We also show that both heterogeneity of transmission and life-history tradeoffs modulate characteristics of population dynamics such as the frequency and intensity of “boom–bust" population cycles. Furthermore, we show that because of life-history tradeoffs affecting the transmission rate, the use of multiple pathogen strains is more effective than the use of a single strain to control insect densities only when the pathogen strains differ considerably in their transmission characteristics. By quantifying the effects of ecology and evolution on population densities, we are able to offer recommendations to assess the long-term effects of classical biocontrol.</p></abstract><kwd-group><kwd>transmission</kwd><kwd>heterogeneity</kwd><kwd>tradeoffs</kwd><kwd>fecundity</kwd><kwd>genetic variation</kwd><kwd>insect pathogens</kwd><kwd>population dynamics</kwd></kwd-group></article-meta></front><floats-group><fig id="viruses-12-00141-f004" orientation="portrait" position="float"><label>Figure 4</label><caption><p>(<bold>a</bold>) Mean insect density (<inline-formula><mml:math id="mm38"><mml:mrow><mml:mrow><mml:mi>l</mml:mi><mml:mi>o</mml:mi><mml:msub><mml:mi>g</mml:mi><mml:mn>10</mml:mn></mml:msub></mml:mrow></mml:mrow></mml:math></inline-formula>) as a function of the mean transmission rate, under different values for heterogeneity in transmission <inline-formula><mml:math id="mm39"><mml:mrow><mml:msup><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></inline-formula> (Model 1). (<bold>b</bold>) Effects of heterogeneity of transmission and life history tradeoffs on the long-term insect population density for four different models. Models are numbered according to <xref rid="viruses-12-00141-t001" ref-type="table">Table 1</xref>; note that Model 0 (linear transmission) is not included in (<bold>b</bold>) as it does not include heterogeneity of transmission. The four models shown here are: “No tradeoff (Model 1)” with mean transmission held constant at <inline-formula><mml:math id="mm40"><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mn>0.7</mml:mn></mml:mrow></mml:mrow></mml:math></inline-formula> [<xref rid="B72-viruses-12-00141" ref-type="bibr">72</xref>]; “Transmission-heterogeneity tradeoff (Model 2)”, with mean transmission <inline-formula><mml:math id="mm41"><mml:mrow><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula> as a non-linear increasing function of heterogeneity <inline-formula><mml:math id="mm42"><mml:mrow><mml:msup><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></inline-formula> [<xref rid="B79-viruses-12-00141" ref-type="bibr">79</xref>]; “Host evolution (Model 3)” in which mean transmission evolves in response to a tradeoff between host susceptibility and fecundity [<xref rid="B15-viruses-12-00141" ref-type="bibr">15</xref>]; and “Two pathogens (Model 4)”, with one pathogen strain subject to the same tradeoff as in Model 2, and the other strain with constant <inline-formula><mml:math id="mm43"><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mn>0.7</mml:mn></mml:mrow></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="mm44"><mml:mrow><mml:mrow><mml:msup><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>0.576</mml:mn></mml:mrow></mml:mrow></mml:math></inline-formula>. The two pathogen strain model is only shown for the range of heterogeneity values over which two-strain coexistence occurs. Models 2 and 4 do not cover the full extent of <inline-formula><mml:math id="mm45"><mml:mrow><mml:msup><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></inline-formula> values because they are specifically constrained by the shape of the tradeoff curve shown in <xref ref-type="boxed-text" rid="viruses-12-00141-box002">Box 2</xref>.</p></caption><graphic xlink:href="viruses-12-00141-g004"></graphic></fig><fig id="viruses-12-00141-f005" orientation="portrait" position="float"><label>Figure 5</label><caption><p>Effects of heterogeneity of transmission and life history tradeoffs on the long-term population dynamics of host insects for four different models. Models are numbered according to <xref rid="viruses-12-00141-t001" ref-type="table">Table 1</xref>; note that Model 0 (linear transmission) is not included as it does not include heterogeneity of transmission. See <xref ref-type="fig" rid="viruses-12-00141-f004">Figure 4</xref>b for details on models and parameter values. For any given value of <inline-formula><mml:math id="mm46"><mml:mrow><mml:msup><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></inline-formula> multiple insect densities (dots) occur whenever the insect population cycles through time. Thus, the dot range across the y-axis indicates the cycle amplitudes, whereas the number of dots indicate the frequency of recurring cycle peaks or troughs averaged over a period of 30 generations. Light orange vertical bars indicate the six heterogeneity values plotted in <xref ref-type="fig" rid="viruses-12-00141-f007">Figure 7</xref>.</p></caption><graphic xlink:href="viruses-12-00141-g005"></graphic></fig><fig id="viruses-12-00141-f006" orientation="portrait" position="float"><label>Figure 6</label><caption><p>Insect density as a function of fecundity, heterogeneity of transmission, and the heritability of transmission as predicted by the evolution model (Model 3) where low transmission is costly to reproduction. For any given value of insect fecundity multiple insect densities (dots) occur whenever the insect population cycles through time. Thus, the dot range across the y-axis indicates the cycle amplitudes, whereas the number of dots indicate the frequency of recurring cycle peaks or troughs averaged over a period of 30 generations.</p></caption><graphic xlink:href="viruses-12-00141-g006"></graphic></fig><fig id="viruses-12-00141-f007" orientation="portrait" position="float"><label>Figure 7</label><caption><p>Host and pathogen population densities for four models (rows) incorporating heterogeneity of transmission <inline-formula><mml:math id="mm47"><mml:mrow><mml:msup><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></inline-formula> (see <xref rid="viruses-12-00141-t001" ref-type="table">Table 1</xref>) under realistic parameter values. See <xref ref-type="fig" rid="viruses-12-00141-f004">Figure 4</xref>b for details on models and parameter values. Heterogeneity <inline-formula><mml:math id="mm48"><mml:mrow><mml:msup><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></inline-formula> is considered for five different cases (columns left to right): Very low heterogeneity (<inline-formula><mml:math id="mm49"><mml:mrow><mml:mrow><mml:msup><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>0.001</mml:mn></mml:mrow></mml:mrow></mml:math></inline-formula>), low heterogeneity (<inline-formula><mml:math id="mm50"><mml:mrow><mml:mrow><mml:msup><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>0.7</mml:mn></mml:mrow></mml:mrow></mml:math></inline-formula>), intermediate heterogeneity (<inline-formula><mml:math id="mm51"><mml:mrow><mml:mrow><mml:msup><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>1.5</mml:mn></mml:mrow></mml:mrow></mml:math></inline-formula>; the minimum value value where two-pathogen coexistence occurs in Model 4), high heterogeneity (<inline-formula><mml:math id="mm52"><mml:mrow><mml:mrow><mml:msup><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>2.97</mml:mn></mml:mrow></mml:mrow></mml:math></inline-formula>, the realistic estimate for the host evolution model (Model 3) see [<xref rid="B15-viruses-12-00141" ref-type="bibr">15</xref>], and very high heterogeneity (<inline-formula><mml:math id="mm53"><mml:mrow><mml:mrow><mml:msup><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:mrow></mml:math></inline-formula>; the maximum heterogeneity considered). Note that the range of possible heterogeneity values is restricted by the transmission-heterogeneity tradeoff function for Models 2 and 4, and thus the lowest and highest cases are missing for these models. Notice also that for near-zero heterogeneity and no tradeoff (upper left corner), neither the host nor the pathogen go extinct, rather, they reach values very close to zero and so eventually the the host population booms.</p></caption><graphic xlink:href="viruses-12-00141-g007"></graphic></fig><fig id="viruses-12-00141-f008" orientation="portrait" position="float"><label>Figure 8</label><caption><p>Effects of varying host survival and fecundity costs of transmission on the average transmission value of the following generation <inline-formula><mml:math id="mm54"><mml:mrow><mml:msub><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>. Here, the change in transmission is relative to the transmission value of the previous generation, <italic>n</italic>, described as <inline-formula><mml:math id="mm55"><mml:mrow><mml:mfrac><mml:mrow><mml:msub><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:msub><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mi>n</mml:mi></mml:msub></mml:mfrac></mml:mrow></mml:math></inline-formula>. The fecundity costs of transmission is given by the expression <inline-formula><mml:math id="mm56"><mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>s</mml:mi><mml:msub><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo>[</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mi>n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>Z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></inline-formula> [<xref rid="B15-viruses-12-00141" ref-type="bibr">15</xref>], where <inline-formula><mml:math id="mm57"><mml:mrow><mml:mrow><mml:mi>I</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mi>n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>Z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> is a function describing host mortality in the previous generation, and is itself a function of the insect and pathogen population densities (<inline-formula><mml:math id="mm58"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="mm59"><mml:mrow><mml:msub><mml:mi>Z</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, respectively) and the transmission rate <inline-formula><mml:math id="mm60"><mml:mrow><mml:msub><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>). This mortality function is proportional to the intensity of selection on transmission [<xref rid="B15-viruses-12-00141" ref-type="bibr">15</xref>]. These results were produced by setting <inline-formula><mml:math id="mm61"><mml:mrow><mml:mrow><mml:msub><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math id="mm62"><mml:mrow><mml:mrow><mml:mi>b</mml:mi><mml:mo>=</mml:mo><mml:mn>0.13</mml:mn></mml:mrow></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math id="mm63"><mml:mrow><mml:mrow><mml:msup><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>2.97</mml:mn></mml:mrow></mml:mrow></mml:math></inline-formula> and by varying <italic>s</italic> and <inline-formula><mml:math id="mm64"><mml:mrow><mml:mrow><mml:mi>I</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mi>n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>Z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> between 0 and 1.</p></caption><graphic xlink:href="viruses-12-00141-g008"></graphic></fig><table-wrap id="viruses-12-00141-t001" orientation="portrait" position="float"><object-id pub-id-type="pii">viruses-12-00141-t001_Table 1</object-id><label>Table 1</label><caption><p>Models used to compare effects of biocontrol on population dynamics and the equilibrium population size of host pest insects. <inline-formula><mml:math id="mm65"><mml:mrow><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula> is mean transmission rate, <italic>C</italic> coefficient of variation of transmission, <inline-formula><mml:math id="mm66"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> is initial density of susceptible hosts, <italic>P</italic> is pathogen density, <italic>b</italic> is heritability, <italic>s</italic> is the slope of fecundity and transmission, and <italic>i</italic> indicates pathogen strain <italic>i</italic>. The function <inline-formula><mml:math id="mm67"><mml:mrow><mml:mrow><mml:mi>I</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mi>n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>Z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> quantifies the fraction of insects killed by virus and is thus a selective force on <inline-formula><mml:math id="mm68"><mml:mrow><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>. Mode numbering begins at zero, as Model 1 converges to the linear transmission model when heterogeneity approaches zero, and thus Model (0) is not referenced later in this work.</p></caption><table frame="hsides" rules="groups"><thead><tr><th align="left" valign="middle" style="border-bottom:solid thin;border-top:solid thin" rowspan="1" colspan="1">Model Description</th><th align="center" valign="middle" style="border-bottom:solid thin;border-top:solid thin" rowspan="1" colspan="1">Force of Infection</th><th align="left" valign="middle" style="border-bottom:solid thin;border-top:solid thin" rowspan="1" colspan="1">Heterogeneity</th><th align="left" valign="middle" style="border-bottom:solid thin;border-top:solid thin" rowspan="1" colspan="1">Tradeoffs</th><th align="center" valign="middle" style="border-bottom:solid thin;border-top:solid thin" rowspan="1" colspan="1">Reference</th></tr></thead><tbody><tr><td align="left" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1">(<bold>0</bold>) One pathogen strain, homogeneous host, no evolution</td><td align="center" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1"><inline-formula><mml:math id="mm69"><mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mi>P</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:mrow></mml:math></inline-formula></td><td align="left" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1">None (linear transmission described by <inline-formula><mml:math id="mm70"><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:math></inline-formula>)</td><td align="left" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1">None</td><td align="center" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1">[<xref rid="B44-viruses-12-00141" ref-type="bibr">44</xref>]</td></tr><tr><td align="left" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1">(<bold>1</bold>) One pathogen strain, Gamma distribution of host susceptibility, no evolution, no tradeoffs</td><td align="center" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1"><inline-formula><mml:math id="mm71"><mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mi>P</mml:mi><mml:msup><mml:mfrac><mml:mi>S</mml:mi><mml:msub><mml:mi>S</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mfrac><mml:msup><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:msup></mml:mrow></mml:mrow></mml:math></inline-formula></td><td align="left" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1">Host susceptibility, described by CV of transmission (<italic>C</italic>)</td><td align="left" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1">None</td><td align="center" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1">[<xref rid="B44-viruses-12-00141" ref-type="bibr">44</xref>,<xref rid="B72-viruses-12-00141" ref-type="bibr">72</xref>]</td></tr><tr><td align="left" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1">(<bold>2</bold>) One pathogen strain, same as (1) above but with evolutionary tradeoff between pathogen mean transmission and heterogeneity of transmission described by function <inline-formula><mml:math id="mm72"><mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1"><inline-formula><mml:math id="mm73"><mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow><mml:mi>P</mml:mi><mml:msup><mml:mfrac><mml:mi>S</mml:mi><mml:msub><mml:mi>S</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mfrac><mml:msup><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:msup></mml:mrow></mml:mrow></mml:math></inline-formula></td><td align="left" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1">Host susceptibility, described by CV of transmission (<italic>C</italic>)</td><td align="left" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1">Evolutionary tradeoff between pathogen mean transmission and heterogeneity of transmission described by function <inline-formula><mml:math id="mm74"><mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1">[<xref rid="B72-viruses-12-00141" ref-type="bibr">72</xref>,<xref rid="B79-viruses-12-00141" ref-type="bibr">79</xref>]</td></tr><tr><td align="left" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1">(<bold>3</bold>) One pathogen strain, Gamma distribution of host susceptibility, host evolution</td><td align="center" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1"><inline-formula><mml:math id="mm75"><mml:mrow><mml:mrow><mml:msub><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mi>n</mml:mi></mml:msub><mml:mi>P</mml:mi><mml:msup><mml:mfrac><mml:mi>S</mml:mi><mml:msub><mml:mi>S</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mfrac><mml:msup><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:msup></mml:mrow></mml:mrow></mml:math></inline-formula></td><td align="left" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1">Host susceptibility described by the average transmission <inline-formula><mml:math id="mm76"><mml:mrow><mml:msub><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> in generation <italic>n</italic> and genetic and environmental variation in susceptibility <inline-formula><mml:math id="mm77"><mml:mrow><mml:mrow><mml:mi>b</mml:mi><mml:msup><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:math></inline-formula></td><td align="left" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1">Selection on <inline-formula><mml:math id="mm78"><mml:mrow><mml:msub><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> imposed through host mortality and a tradeoff between host reproduction and susceptibility, described by <inline-formula><mml:math id="mm79"><mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>s</mml:mi><mml:msub><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo>[</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mi>n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>Z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1">[<xref rid="B15-viruses-12-00141" ref-type="bibr">15</xref>]</td></tr><tr><td align="left" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1">(<bold>4</bold>) Two pathogen strains, Gamma distribution of host susceptibility, no evolution</td><td align="center" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1"><inline-formula><mml:math id="mm80"><mml:mrow><mml:mrow><mml:msub><mml:mi>ν</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow><mml:mi>S</mml:mi><mml:msub><mml:mi>P</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:mrow></mml:math></inline-formula></td><td align="left" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1">Host susceptibility and pathogen genotype, described by mean <inline-formula><mml:math id="mm81"><mml:mrow><mml:msub><mml:mi>ν</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, CV of transmission <inline-formula><mml:math id="mm82"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, and interstrain transmission correlation <inline-formula><mml:math id="mm83"><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow></mml:math></inline-formula></td><td align="left" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1">Evolutionary tradeoff between pathogen mean transmission and heterogeneity of transmission described by function <inline-formula><mml:math id="mm84"><mml:mrow><mml:mrow><mml:msub><mml:mi>ν</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></inline-formula></td><td align="center" valign="middle" style="border-bottom:solid thin" rowspan="1" colspan="1">[<xref rid="B79-viruses-12-00141" ref-type="bibr">79</xref>]</td></tr></tbody></table></table-wrap></floats-group></pmc></record>Pour manipuler ce document sous Unix (Dilib)
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