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Piggyback-the Winner (PtW)
Unlike KtW, the Piggyback-the-Winner (PtW) hypothesis makes a key distinction between temperate and lytic viruses. PtW posits that temperate viruses are more abundant than lytic viruses at high host density and growth rate. Hosts may actively recruit temperate viruses, which protect the host from closely related viruses, a phenomenon known as super-infection exclusion. Abundant hosts are parasitized, rather than killed, and high host densities are maintained.

Coevolving KtW Model (CKTW)
One problematic assumption of the KtW hypothesis is that predator and prey populations are infinite or very large. Given this assumption, the Lotka-Volterra equations would unrealistically imply that neither predators nor prey would go extinct. A 2017 study by Chi Xue and Nigel Goldenfeld tested the effects of the original KtW model on finite populations. Introducing stochasticity resulted in successive extinctions, showing that species coexistence could not be maintained. However, KtW models can be adapted to allow predators and prey to coevolve by mutation. As prey evolve to evade predators and predators evolve to overcome new defenses, new mutants are introduced. Thus, Coevolving KtW (CKTW) models allow species diversity to be maintained, even for finite populations.

Article

- Original KTW model as deterministic

- Assumptions (spatially homogenous, large # of individuals)

- Avoid cascade of extinctions by allowing pred + prey to coevolve by mutation

Quanta

- Only one species should occupy an ecological niche ==> competitive exclusion

- Role of predator-prey relationships in ecosystems (predation pushes number of prey back down) ==> combination of predation and competition

- KtW issue ==> describe populations as if individuals did not exist?

- Models underestimating extinction? ==> lack of stochastic noise?

- Every species goes to extinction when you add noise? (no diversity)

- introduce idea of evolution

- Selection pressure to resist viruses exceeds other pressures

- Viruses as predator and taxi driver for genes

- issues with coevolving KtW model (assumes evolutionary and ecological changes happen on the same timescale / same frequency)

The reason that the stable deterministic steady state of the generalized KTW model cannot be maintained in the presence of demographic stochasticity lies in the fact that species populations in the stochastic model are all finite, and the probability of the population reaching zero due to a random fluctuation is always nonzero.

'''Prey and predator coevolve with each other so that fit mutants are constantly being introduced into the system, thus preventing the elimination of the species. Specifically, prey improve their phenotypic traits (e.g., strengthening the shell) to escape from predators, and predators also adjust their corresponding traits (e.g., sharpening the claws) to catch prey. This coevolutionary arms race has been well documented in many systems [33,34]. Previous theoretical studies focused on the dynamics of the traits of prey and predator groups [35–37], and the structure of the predation network [38], under different coevolving modes. Here, we study how coevolution affects the diversity of the hostspecific predation system.'''

[explain contrast from Ktw]

- Find images for alternative hypotheses?

Current understanding on KtW stems from our knowledge of lytic viruses and their host populations.

'''Piggyback-the-Winner (PTW) is a similar dynamic model of bacteria-virus interactions but incorporates the viral life cycle into the model. PTW model states that the nonlinear relationship between viruses and prokaryotes is observed due to viral dynamics being suppressed at high host densities and super-infection exclusion, rather than developed “resistance” as suggested by the KtW model.'''