# Classification of NMS-flows with unique twisted saddle orbit on orientable
4-manifolds

Vladislav Galkin111ORCID: 0000-0001-6796-9228 National Research University
Higher School of Economics; 25/12, Bolshaya Pecherskaya St., Nizhny Novgorod,
Russia, 603155  Saint Petersburg University, 7/9 Universitetskaya nab., St.
Petersburg, 199034 Russia  Olga Pochinka222ORCID: 0000-0002-6587-5305 National
Research University Higher School of Economics; 25/12, Bolshaya Pecherskaya
St., Nizhny Novgorod, Russia, 603155  Saint Petersburg University, 7/9
Universitetskaya nab., St. Petersburg, 199034 Russia  Danila Shubin333ORCID:
0000-0002-8495-4826; Corresponding author<EMAIL_ADDRESS>National
Research University Higher School of Economics; 25/12, Bolshaya Pecherskaya
St., Nizhny Novgorod, Russia, 603155  Saint Petersburg University, 7/9
Universitetskaya nab., St. Petersburg, 199034 Russia

###### Abstract

Topological equivalence of Morse-Smale flows without fixed points (NMS-flows)
under assumptions of different generalities was studied in a number of papers.
In some cases when the number of periodic orbits is small, it is possible to
give exhaustive classification, namely to provide the list of all manifolds
that admit flows of considered class, find complete invariant for topological
equivalence and introduce each equivalence class with some representative
flow. This work continues the series of such articles. We consider the class
of NMS-flows with unique saddle orbit, under the assumption that it is
twisted, on closed orientable 4-manifolds and prove that the only 4-manifold
admitting the considered flows is the manifold
$\mathbb{S}^{3}\times\mathbb{S}^{1}$. Also, it is established that such flows
are split into exactly eight equivalence classes and construction of a
representative for each equivalence class is provided.

##  1 Introduction and main results

In the present paper we consider _NMS-flows_ $f^{t}$, namely _non-singular_
(without fixed points) Morse-Smale flows which are defined on orientable-
manifold $M^{4}$. Non-wandering set of such flow consist of a finite number of
hyperbolic periodic orbits. Asimov proved [1] that ambient manifold of such
flow is a union of round handles. However, if the number of orbits is small,
topology of the ambient manifold can be specified. For instance, only lens
spaces admit NMS-flows with two periodic orbits. Moreover, it was shown in [9]
that each lens space admit exactly two classes of topological equivalence
except for 3-sphere $\mathbb{S}^{3}$ and projective space
$\mathbb{R}\text{P}^{3}$ which both admit the unique equivalence class.

Campos et al. [3] argued that lens spaces are the only prime 3-manifolds that
are ambient for NMS-flow with unique saddle periodic orbit, but this is not
so. There exists infinite series of mapping tori non-homeomorphic to lens
spaces which admit such flows [13]. Moreover, necessary and sufficient
conditions for topological equivalence of such flows were obtained in [10].
Finally, any 3-manifold admitting such flows is a lens space or connected sum
of two lens spaces or a small Seifert fibered 3-manifold [11]. Note, that
these results cannot be deduced from the classification of Morse-Smale flows
with finite number of singular trajectories made by Umanskii [15].

In the present paper we establish the topology of orientable 4-manifolds that
are ambient for NMS-flows with exactly one saddle orbit assuming that it is
_twisted_ (its invariant manifolds are non-orientable). Besides, the number of
equivalence classes of such flows on each admissible manifold is calculated.

Let us proceed to the formulation of the results.

Let $M^{4}$ be connected closed orientable 4-manifold, $f^{t}\colon M^{4}\to
M^{4}$ be NMS-flow and $\mathcal{O}$ be its periodic orbit. There exists
tubular neighborhood $V_{\mathcal{O}}$ homeomorphic to
$\mathbb{D}^{3}\times\mathbb{S}^{1}$ such that the flow is topologically
equivalent to the suspension over a linear diffeomorphism of the plane defined
by the matrix which determinant is positive and eigenvalues are different from
$\pm 1$ (see. Proposition 1). If absolute values of both eigenvalues are
greater (less) than one, the corresponding periodic orbit is _attracting
(repelling)_ , otherwise it is _saddle_. The saddle orbit is called _twisted_
if both eigenvalues are negative and _non-twisted_ otherwise.

Consider the class $G^{-}_{3}(M^{4})$ of NMS-flows $f^{t}\colon M^{4}\to
M^{4}$ with unique saddle orbit which is twisted. Since the ambient manifold
$M^{4}$ is the union of stable (unstable) manifolds of its periodic orbits,
the flow $f^{t}\in G^{-}_{3}(M^{4})$ has at least one attracting and at least
one repelling orbit. In Section 2 the following fact is established.

######  Lemma 1.

The non-wandering set of any flow $f^{t}\in G^{-}_{3}(M^{4})$ consists of
exactly three periodic orbits $S,A,R$, saddle, attracting and repelling,
respectively.

Saddle orbit $S$ of the flow $f^{t}\in G^{-}_{3}(M^{4})$ can be either 3- or
2-dimensional; let $G^{-1}_{3}(M^{4}),\ G^{-2}_{3}(M^{4})$ denote
corresponding subclasses of $G^{-}_{3}(M^{4})$. Obviously, since dimension of
unstable manifold is invariant under an equivalence homeomorphism no flow in
$G^{-1}_{3}(M^{4})$ is topologically equivalent to any flow in
$G^{-2}_{3}(M^{4})$. Furthermore, $G^{-2}_{3}(M^{4})=\\{f^{-t}\colon f^{t}\in
G^{-1}_{3}(M^{4})\\}$ and the flows $f^{t},f^{\prime t}$ are topologically
equivalent if and only if $f^{-t},f^{\prime-t}$ are topologically equivalent.
This immediately implies that classification in the class $G^{-}_{3}(M^{4})$
reduces to classification in the subclass $G^{-1}_{3}(M^{4})$.

Let $f^{t}\in G^{-1}_{3}(M^{4})$. Since the flow $f^{t}$ in some tubular
neighborhood of is topologically equivalent to the suspension over linear
diffeomorphism of the plane, the topology of periodic orbits $A,S,R$ stable
and unstable manifolds is:

  * •

$W^{u}_{S}\cong\mathbb{R}^{2}\tilde{\times}\mathbb{S}^{1}$ (open solid Klein
bottle);

  * •

$W^{s}_{S}\cong\mathbb{R}\tilde{\times}\mathbb{S}^{1}$ (open Möbius band);

  * •

$W^{s}_{A}\cong W^{u}_{R}\cong\mathbb{R}^{2}\times\mathbb{S}^{1}$ (open solid
torus);

  * •

$W^{u}_{A}\cong W^{s}_{R}\cong\mathbb{S}^{1}$ (circle).

Let $\mathcal{O}\in\\{S,A,R\\}$. Choose the generator
$\mathcal{G}_{\mathcal{O}}$ of boundary $T_{\mathcal{O}}=\partial
V_{\mathcal{O}}\cong\mathbb{S}^{2}\times\mathbb{S}^{1}$ fundamental group
which is homologous to $\mathcal{O}$ in
$V_{\mathcal{O}}\cong\mathbb{D}^{3}\times\mathbb{S}^{1}$. By definition the
manifold $T_{S}$ is secant for all flow trajectories except the periodic ones.
Since the flow in some tubular neighborhood of $S$ is topologically equivalent
to suspension, the set $K_{S}=W^{u}_{S}\cap T_{S}$ is homeomorphic to the
Klein bottle. Let $\lambda_{S},\mu_{S}$ be the knots (simple closed curves),
which are generators of the fundamental group $\pi_{1}(K_{S})$ with relation
$[\lambda_{S}*\mu_{S}]=[\mu_{S}^{-1}*\lambda_{S}]$. We will call the curve
$\mu_{S}$ _meridian_ and the curve $\lambda_{S}$ _longitude_. By virtue of
Proposition 3 the Klein bottle longitude embedded in
$\mathbb{S}^{2}\times\mathbb{S}^{1}$ is a generator of fundamental group
$\pi_{1}(\mathbb{S}^{2}\times\mathbb{S}^{1})$. Consider the longitude
$\lambda_{S}$ be oriented in such way that its homotopy type
$\langle\lambda_{S}\rangle$ in $T_{S}$ coincide with type
$\langle\mathcal{G}{{}_{S}}\rangle$. So, the set $K_{A}=W^{u}_{S}\cap T_{A}$
is the Klein bottle with longitude $\lambda_{A}$ which is pointwise
transferred along the flow $f^{t}$ orbits from $\lambda_{S}$.

Since the flow in some tubular neighborhood of $S$ is topologically equivalent
to suspension the set $\gamma_{S}=W^{s}_{S}\cap T_{S}$ is a knot in $T_{S}$,
wrapping around $\mathcal{G}_{S}$ twice. We will assume that the knot
$\gamma_{S}$ is oriented in such way that its homotopy type
$\langle\gamma_{S}\rangle$ on $T_{S}$ coincides with homotopy type of
$2\langle\mathcal{G}{{}_{S}}\rangle$. So the set $\gamma_{R}=W^{s}_{S}\cap
T_{R}$ is a knot in $T_{R}$ and its orientation is induced by the flow $f^{t}$
from $\gamma_{S}$.

######  Lemma 2.

Let $f^{t}\in G^{-1}_{3}(M^{4})$ then the following conditions hold:

  1. 1.

$\langle\lambda_{A}\rangle=\delta_{A}\langle\mathcal{G}_{A}\rangle,\,\delta_{A}\in\\{-1,+1\\}$
in $T_{A}$;

  2. 2.

$\langle\gamma_{R}\rangle=\delta_{R}\langle\mathcal{G}_{R}\rangle,\,\delta_{R}\in\\{-1,+1\\}$
in $T_{R}$.

Let

$C_{f^{t}}=(\delta_{A},\delta_{R}).$

######  Theorem 1.

Flows $f^{t},\,f^{\prime t}\in G^{-1}_{3}(M^{4})$ are topologically equivalent
if and only if $C_{f^{t}}=C_{f^{\prime t}}$.

######  Theorem 2.

For any element $C\in\mathbb{S}^{0}\times\mathbb{S}^{0}$ there exists a flow
$f^{t}\in G_{3}^{-1}(M^{4})$ such that $C=C_{f^{t}}$.

######  Theorem 3.

The only 4-manifold that is ambient for a flow of the class $G^{-}_{3}(M^{4})$
is $\mathbb{S}^{3}\times\mathbb{S}^{1}$. Moreover
$G^{-}_{3}(\mathbb{S}^{3}\times\mathbb{S}^{1})$ consists of eight classes of
topological equivalence.

Note that weakening the saddle orbit twistedness condition fundamentally
changes the picture. For example, in [8] non-singular flows that are
suspensions over Pixton diffeomorphisms on a three-dimensional sphere are
considered. It is proved that in the class under consideration there exist
flows with wildly embedded invariant manifolds of the saddle orbit. Moreover,
there are an infinite number of topological equivalence classes for such
flows.

Acknowledgements.  This work was performed at the Saint Petersburg Leonhard
Euler International Mathematical Institute and supported by the Ministry of
Science and Higher Education of the Russian Federation (agreement no.
075-15-2022-287).

##  2 Flows of the class $G^{-1}_{3}(M^{4})$

###  2.1 Structure of periodic orbits

This section is devoted to proof of Lemma 1: non-wandering set of any flow
$f^{t}\in G^{-1}_{3}(M^{4})$ consists of three periodic orbits $S,A,R$,
saddle, attracting and repelling respectively.

###### Proof.

The proof is based on the following representation of the ambient manifold
$M^{4}$ of the NMS-flow $f^{t}$ with the set of periodic orbits $Per_{f^{t}}$
(see, for example, [14])

$M^{4}=\bigcup\limits_{\mathcal{O}\in
Per_{f^{t}}}W^{u}_{\mathcal{O}}=\bigcup\limits_{\mathcal{O}\in
Per_{f^{t}}}W^{s}_{\mathcal{O}},$ (1)

as well as the asymptotic behavior of invariant manifolds

${\rm cl}(W^{u}_{\mathcal{O}})\setminus
W^{u}_{\mathcal{O}}=\bigcup\limits_{\tilde{\mathcal{O}}\in Per_{f^{t}}\colon
W^{u}_{\mathcal{O}}\cap
W^{s}_{\mathcal{O}}\neq\varnothing}W^{u}_{\tilde{\mathcal{O}}},$ (2) ${\rm
cl}(W^{s}_{\mathcal{O}})\setminus
W^{s}_{\mathcal{O}}=\bigcup\limits_{\tilde{\mathcal{O}}\in Per_{f^{t}}\colon
W^{s}_{\mathcal{O}}\cap
W^{u}_{\mathcal{O}}\neq\varnothing}W^{s}_{\tilde{\mathcal{O}}}.$ (3)

In particular, it follows from eq. 1 that any NMS-flow has at least one
attracting orbit and at least one repulsive one. Moreover, if an NMS-flow has
a saddle periodic orbit, then the basin of any attracting orbit has a non-
empty intersection with an unstable manifold of at least one saddle orbit (see
Proposition 2.1.3 [4]) and a similar situation with the basin of a repulsive
orbits.

Now let $f^{t}\in G_{3}^{-1}(M^{3})$ and $S$ be its only saddle orbit. It
follows from the relation (1) that $W^{u}_{S}\setminus S$ intersects only
basins of attracting orbits. Since the set $W^{u}_{S}\setminus S$ is connected
and the basins of attracting orbits are open, then $W^{u}_{S}$ intersects
exactly one such basin. Denote by $A$ the corresponding attracting orbit.
Since there is only one saddle orbit, there is only one attracting orbit.
Similar reasoning for $W^{s}_{S}$ leads to the existence of a unique repulsive
orbit $R$. ∎

###  2.2 Canonical neighborhoods of periodic orbits

Recall the definition of a superstructure. Let $\varphi\colon M^{3}\to M^{3}$
be a diffeomorphism of a 3-manifold. We define the diffeomorphism
$g_{\varphi}\colon M^{3}\times\mathbb{R}^{1}\to M^{3}\times\mathbb{R}^{1}$ by
the formula

$g_{\varphi}(x_{1},x_{2},x_{3},x_{4})=(\varphi(x_{1},x_{2},x_{3}),x_{4}-1).$

Then the group $\\{g_{\varphi}^{n}\\}\cong\mathbb{Z}$ acts freely and
discontinuously on $M^{3}\times\mathbb{R}^{1}$, whence the orbit space
$\Pi_{\varphi}=M^{3}\times\mathbb{R}^{1}g_{\varphi}$ is a smooth 4-manifold,
and the natural projection $v_{\varphi}\colon
M^{3}\times\mathbb{R}^{1}\to\Pi_{\varphi}$ is a covering. At the same time,
the flow $\xi^{t}\colon M^{3}\times\mathbb{R}^{1}\to
M^{3}\times\mathbb{R}^{1}$, given by the formula

$\xi^{t}(x_{1},x_{2},x_{3},x_{4})=(x_{1},x_{2},x_{3},x_{4}+t),$

induces the flow
$[\varphi]^{t}=v_{\varphi}\xi^{t}v^{-1}_{\varphi}\colon\Pi_{\varphi}\to\Pi_{\varphi}$.
The flow $[\varphi]^{t}$ is called the _suspension of the diffeomorphism
$\varphi$_.

We define the diffeomorphisms
$a_{0},a_{1},a_{2},a_{3}\colon\mathbb{R}^{3}\to\mathbb{R}^{3}$ by the formulas

$a_{3}(x_{1},x_{2},x_{3})=(2x_{1},2x_{2},2x_{3}),\ a_{0}=a_{3}^{-1},$ $a_{\pm
1}(x_{1},x_{2},x_{3})=(\pm 2x_{1},\pm 1/2x_{2},1/2x_{3}),\ a_{\pm 2}=a_{\pm
1}^{-1}.$

Let

$\displaystyle V_{0}=\\{(x_{1},x_{2},x_{3},x_{4})\in\mathbb{R}^{3}|\
4^{x_{4}}x_{1}^{2}+4^{x_{4}}x^{2}_{2}+4^{x_{4}}x^{2}_{3}\leqslant 1\\},$
$\displaystyle V_{\pm 1}=\\{(x_{1},x_{2},x_{3},x_{4})\in\mathbb{R}^{3}|\
4^{-x_{4}}x_{1}^{2}+4^{x_{4}}x^{2}_{2}+4^{x_{4}}x^{2}_{3}\leqslant 1\\},$
$\displaystyle V_{\pm 2}=\\{(x_{1},x_{2},x_{3},x_{4})\in\mathbb{R}^{3}|\
4^{-x_{4}}x_{1}^{2}+4^{-x_{4}}x^{2}_{2}+4^{x_{4}}x^{2}_{3}\leqslant 1\\},$
$\displaystyle V_{3}=\\{(x_{1},x_{2},x_{3},x_{4})\in\mathbb{R}^{3}|\
4^{-x_{4}}x_{1}^{2}+4^{-x_{4}}x^{2}_{2}+4^{-x_{4}}x^{2}_{3}\leqslant 1\\}.$

For $i\in\\{0,\pm 1,\pm 2,3\\}$ we set $v_{i}=v_{a_{i}}$, $T_{i}=\partial
V_{i}$ and $\mathbb{V}_{i}=v_{i}(V_{i}),\ \mathbb{T}_{i}=v_{i}(T_{i})$.

The following statement, proved by M. Irwin [5], describes the behavior of
flows in a neighborhood of hyperbolic periodic orbits.

######  Proposition 1 (M. Irwin [5]).

If $\mathcal{O}$ is a hyperbolic orbit of a flow $f^{t}\colon M^{4}\to M^{4}$
defined on an orientable 4-manifold $M^{4}$, then there exists a tubular
neighborhood $V_{\mathcal{O}}$ of the orbit $\mathcal{O}$ such that the flow
$f^{t}\big{|}_{V_{\mathcal{O}}}$ is topologically equivalent, by means of some
homeomorphism $H_{\mathcal{O}}$, to one of the following streams:

  * •

$[a_{0}]^{t}|_{\mathbb{V}_{0}}$ if $\mathcal{O}$ is an attracting orbit;

  * •

$[a_{1}]^{t}|_{\mathbb{V}_{1}}$ if $\mathcal{O}$ is a non-twisted saddle orbit
with a two-dimensional unstable manifold $W^{u}_{\mathcal{O}}$;

  * •

$[a_{-1}]^{t}|_{\mathbb{V}_{-1}}$ if $\mathcal{O}$ is a twisted saddle orbit
with a two-dimensional unstable manifold $W^{u}_{\mathcal{O}}$ ;

  * •

$[a_{2}]^{t}|_{\mathbb{V}_{2}}$ if $\mathcal{O}$ is a non-twisted saddle orbit
with an unstable 3-manifold $W^{u}_{\mathcal{O}}$;

  * •

$[a_{-2}]^{t}|_{\mathbb{V}_{-2}}$ if $\mathcal{O}$ is a twisted saddle orbit
with an unstable 3-manifold $W^{u}_{\mathcal{O}}$ ;

  * •

$[a_{3}]^{t}|_{\mathbb{V}_{3}}$ if $\mathcal{O}$ is a repelling orbit.

The neighborhood
$V_{\mathcal{O}}=H_{\mathcal{O}}(\mathbb{V}_{i}),\,i\in\\{0,\pm 1,\pm 2,3\\}$
described in Proposition 1 is called _canonical neighborhood_ of the periodic
orbit $\mathcal{O}$.

When proving topological equivalence, we will use the following fact, which
follows from the proof of Theorem 4 and Lemma 4 in [9], and can also be found
in [15] (Theorem 1.1).

######  Proposition 2.

A homeomorphism $h_{i}\colon\partial\mathbb{V}_{i}\to\partial\mathbb{V}_{i}$
for $i\in\\{0,3\\}$ extends to a homeomorphism
$H_{i}\colon\mathbb{V}_{i}\to\mathbb{V}_{i}$ realizing the equivalence flow
$[a_{i}]^{t}$ with itself if and only if the induced isomorphism
$h_{i*}\colon\pi_{1}(\partial\mathbb{V}_{i})\to\pi_{1}(\partial\mathbb{V}_{i})$
is identical.

###  2.3 Trajectory mappings

Consider a flow $f^{t}\colon M^{4}\to M^{4}$ from the set $G^{-1}_{3}(M^{3})$.
Then
$V_{A}=H_{A}(\mathbb{V}_{0}),\,V_{R}=H_{R}(\mathbb{V}_{3}),\,V_{S}=H_{S}(\mathbb{V}_{-2})$.
Let $\Gamma=\\{(x_{1},x_{2},x_{3},x_{4})\in T_{-2}|\
4^{x_{4}}x_{3}^{2}=1/2\\},\ \Gamma^{u}=Ox_{2}x_{3}x_{4}\cap T_{-2},\
\Gamma^{s}=Ox_{1}x_{4}\cap T_{-2}$. By construction, the set $T_{-2}$ is
homeomorphic to $\mathbb{S}^{2}\times\mathbb{R}$, the set $\Gamma$ consists of
two surfaces, each of which is homeomorphic to
$\mathbb{S}^{1}\times\mathbb{R}$ , dividing $T_{-2}$ into three connected
components, one of which $N^{u}$ contains the cylinder
$\Gamma^{u}\cong\mathbb{S}^{1}\times\mathbb{R}$, and the union $N^{s}$ the
other two contains a pair of $\Gamma^{s}\cong\mathbb{S}^{0}\times\mathbb{R}$
curves, one curve in each component. Then on $T_{S}$

  * •

$K_{S}=H_{S}(v_{-2}(\Gamma^{u})))$ is a Klein bottle;

  * •

$\gamma_{S}=H_{S}(v_{-2}(\Gamma^{s})))$ is a knot winding twice around the
generator $\mathcal{G}_{S}$;

  * •

$N^{u}_{S}=H_{S}(v_{-2}({\rm cl}(N^{u})))$ is a tubular neighborhood of
$K_{S}$;

  * •

$N^{s}_{S}=H_{S}(v_{-2}({\rm cl}(N^{s})))$ is a tubular neighborhood of
$\gamma_{S}$;

  * •

$\partial N^{u}_{S}=\partial N^{s}_{S}=H_{S}(v_{-2}((\Gamma)))$ is a two-
dimensional torus.

Let

$N^{s}_{R}=\left(\bigcup\limits_{t>0,\,w\in N^{s}_{S}}f^{-t}(w)\right)\cap
T_{R},\,\,N^{u}_{R}=T_{R}\setminus N^{s}_{R},$
$N^{u}_{A}=\left(\bigcup\limits_{t>0,\,w\in N^{u}_{S}}f^{t}(w)\right)\cap
T_{A},\,\,N^{s}_{A}=T_{A}\setminus N^{u}_{A}$

and introduce the following mappings:

  * •

define a continuous function $\tau_{R}\colon T_{R}\to\mathbb{R}^{+}$ so that
$f^{\tau_{R}(r)}(r)\in N^{s}_{S}$ for $r\ inN^{s}_{R}$ and the set
$\mathcal{T}=\bigcup\limits_{r\in N^{u}_{R}}f^{\tau_{{R}}(r)}(r)$ does not
intersect the torus $T_{A}$, we set ${\mathcal{T}}_{R}={\mathcal{T}}\cup
N^{s}_{S}$ and define a homeomorphism $\psi_{R}\colon
T_{R}\to{\mathcal{T}}_{R}$ by the formula $\psi_{R}(r)=f^{\tau_{{R}}(r)}(r)$,
denote by $\mathcal{V}_{R}$ the closure of the connected component of the set
$M^{4}\setminus\mathcal{T}_{R}$ , containing $R$;

  * •

we set ${\mathcal{T}}_{A}={\mathcal{T}}\cup N^{u}_{S}$, define a continuous
function $\tau_{A}:T_{A}\to\mathbb{R}^{+}$ so that
$f^{-\tau_{A}(a)}(a)\in{\mathcal{T}}_{A}$ for $a\in T_{A}$ and define a
homeomorphism $\psi_{A}:T_{A}\to{\mathcal{T}}_{A}$ by the formula
$\psi_{A}(a)=f^{-\tau_{{A}}(a)}(a)$, denote by $\mathcal{V}_{A}$ the closure
of the connected component of the set $M^{4}\setminus\mathcal{T}_{A}$
containing $A$.

We will call the introduced homeomorphisms $\psi_{R},\ \psi_{A}$ _trajectory
maps_. Note that the ambient manifold $M^{4}$ is represented as

$M^{4}=\mathcal{V}_{A}\cup V_{S}\cup\mathcal{V}_{R}.$

Moreover, in the manifolds $\mathcal{V}_{A}$ and $\mathcal{V}_{R}$ the flow
$f^{t}$ is topologically equivalent to the suspensions $[a_{0}]^{t}$ and
$[a_{3}]^{t}$, respectively.

##  3 Homotopy types of knots $\lambda_{A},\gamma_{R}$

In this section, we will prove Lemma 2. To do this, we first describe the
properties of the embedding of the Klein bottle into the manifold
$\mathbb{S}^{2}\times\mathbb{S}^{1}$.

Recall that the Klein bottle $\mathbb{K}$ is the square $[0,1]\times[0,1]$
with sides glued by the relation

$(x,0)\sim(x,1),\quad(0,y)\sim(0,1-y).$

Let $v\colon[0,1]\times[0,1]\to\mathbb{K}$ be the natural projection, then the
curves

$\lambda=v([0,1]\times\\{1/2\\}),\,\mu=v(\\{0\\}\times[0,1])$

are generators of the fundamental group $\pi_{1}(\mathbb{K})$ with relation

$[\lambda*\mu]=[\mu^{-1}*\lambda],$

where the curve $\lambda$ is called _longitude_ and the curve $\mu$ is called
_meridian_.

It is well known that the Klein bottle does not embed into $\mathbb{R}^{3}$,
however, it can be embeddable into $\mathbb{S}^{2}\times\mathbb{S}^{1}$, for
example by defining the embedding
$\tilde{e}_{0}\colon[0,1]\times[0,1]\to\mathbb{S}^{2}\times\mathbb{S}^{1}$ by
the formula

$\tilde{e}_{0}(x,y)=\left(\sin\pi x\cos 2\pi y,\ \cos\pi x\cos 2\pi y,\ \sin
2\pi y,e^{2\pi ix}\right)$

and noticing that
$\tilde{e}_{0}(x,y)=\tilde{e}_{0}(x^{\prime},y^{\prime})\iff(x,y)\sim(x^{\prime},y^{\prime})$.
Then (see, for example, [6, Chapter 5])

$e_{0}=\tilde{e}_{0}v^{-1}\colon\mathbb{K}\to\mathbb{S}^{2}\times\mathbb{S}^{1}$

is the desired embedding of the Klein bottle in
$\mathbb{S}^{2}\times\mathbb{S}^{1}$. Let

$K_{0}=e_{0}(\mathbb{K}).$

Figure 1: Bottle of Klein in $\mathbb{S}^{2}\times\mathbb{S}^{1}$

######  Proposition 3 (Proposition 1.4, [2]).

Let $e\colon\mathbb{K}\to\mathbb{S}^{2}\times\mathbb{S}^{1}$ be an embedding
Klein bottles $\mathbb{K}$, $K=e(\mathbb{K})$,
$N(K)\subset\mathbb{S}^{2}\times\mathbb{S}^{1}$ be a tubular neighborhood of
$K$ and $V(K)=\mathbb{S}^{2}\times\mathbb{S}^{1}\setminus{\rm int}\,N(K)$ (see
Figure 1). Then:

  * 1)

the curve $e(\lambda)$ is a generator of the fundamental group
$\pi_{1}(\mathbb{S}^{2}\times\mathbb{S}^{1})$;

  * 2)

the set $V(K)$ is a solid torus whose meridian is homotopic to the curve
$e(\mu)$;

  * 3)

there exists an orientation-preserving homeomorphism
$h\colon\mathbb{S}^{2}\times\mathbb{S}^{1}\to\mathbb{S}^{2}\times\mathbb{S}^{1}$
such that $h(K)=K_{0}$ and
$h_{*}=id\colon\pi_{1}(\mathbb{S}^{2}\times\mathbb{S}^{1})\to\pi_{1}(\mathbb{S}^{2}\times\mathbb{S}^{1})$.

It remains to prove Lemma 2. To do this, recall that we have chosen
$T_{\mathcal{O}}=\partial
V_{\mathcal{O}}\cong\mathbb{S}^{2}\times\mathbb{S}^{1},\,\mathcal{O}\in\\{A,S,R\\}$
generator $\mathcal{G}_{\mathcal{O}}$ of the fundamental group of
$T_{\mathcal{O}}$, homologous in
$V_{\mathcal{O}}\cong\mathbb{D}^{3}\times\mathbb{S}^{1}$ orbit $\mathcal{O}$.
Due to the fact that canonical neighborhoods of periodic orbits can be chosen
so that $M^{4}=\mathcal{V}_{A}\cup V_{S}\cup\mathcal{V}_{R}$ (see Section
2.3), everywhere below we assume that $V_{A}=\mathcal{V}_{A},\
V_{R}=\mathcal{V}_{R}$.

We also established that the set $K_{S}=W^{u}_{S}\cap T_{S}$ is a Klein bottle
on $T_{S}\cong\mathbb{S}^{2}\times\mathbb{S}^{1}$ and oriented its parallel
$\lambda_{S}$ so that
$\langle\lambda_{S}\rangle=\langle\mathcal{G}_{S}\rangle$ on $T_{S}$. Since
the set $K_{A}=W^{u}_{S}\cap T_{A}$ coincides with $K_{S}$, then
$\lambda_{A}=\lambda_{S}$. We also established that the set
$\gamma_{S}=W^{s}_{S}\cap T_{S}$ is a knot on $T_{S}$ and oriented so that
$\langle\gamma_{S}\rangle=2\langle\mathcal{G}_{S}\rangle$ to $T_{S}$. Since
the set $\gamma_{R}=W^{s}_{S}\cap T_{R}$ coincides with $\gamma_{S}$, then
$\gamma_{R}=\gamma_{S}$.

Let us show that the knots $\lambda_{A},\gamma_{R}$ are generators in the
fundamental groups of the manifolds $T_{A},T_{R}$, respectively.

###### Proof.

Since $T_{A}$ is homeomorphic to the manifold
$\mathbb{S}^{2}\times\mathbb{S}^{1}$ and $\lambda_{A}$ is a parallel of the
Klein bottle $K_{A}\subset T_{A}$, then item 1) of Proposition 3 implies that
$\lambda_{A}$ is a generator in the fundamental group of $T_{A}$, that is, on
$T_{A}$

$\langle\lambda_{A}\rangle=\delta_{A}\langle\mathcal{G}_{A}\rangle$

for $\delta_{A}\in\\{-1,+1\\}$. The set $N(K_{A})=N^{u}_{S}$ is a tubular
neighborhood of the Klein bottle $K_{A}$ in $T_{A}$. It follows from item 2)
of Proposition 3 that the set $V(K_{A})=T_{A}\setminus{\rm int}\,N(K_{A})$ is
a solid torus. The set $N(\gamma_{R})=N^{s}_{S}$ is a tubular neighborhood of
the knot $\gamma_{R}$ in $T_{R}$. On the other hand $T_{R}\setminus{\rm
int}\,N(\gamma_{R})=V(K_{A})$. Thus, the complement to the tubular
neighborhood of $\gamma_{R}$ in $T_{R}$ is a solid torus. By [4, Proposition
4.2], $\gamma_{R}$ is a generator in $T_{R}$ and, therefore,

$\langle\gamma_{R}\rangle=\delta_{R}\langle\mathcal{G}_{R}\rangle$

for $\delta_{R}\in\\{-1,+1\\}$. ∎

##  4 Classification of flows of the set $G^{-1}_{3}(M^{4})$

In this section, we will prove Theorem 1.

###### Proof.

Necessity. Let flows $f^{t}$ and $f^{\prime t}$ have invariants
$C_{f^{t}}=(\delta_{A},\delta_{R}),\ C_{f^{\prime
t}}=(\delta_{A^{\prime}},\delta_{R^{\prime}})$ and are topologically
equivalent by the homeomorphism $H\colon M^{4}\to M^{4}$. Let us show that
$C_{f^{t}}=C_{f^{\prime t}}$.

Let $h_{A}=H\big{|}_{T_{A}}$ and $T_{A^{\prime}}=h(T_{A})$. Then by
Proposition 2

$\langle\mathcal{G}_{A^{\prime}}\rangle=h_{A*}\langle\mathcal{G}_{A}\rangle.$

It follows from item 3) of Proposition 3 that
$\lambda_{{A^{\prime}}}=h_{A}(\lambda_{A})$ is the parallel of the Klein
bottle $K_{A^{\prime}}$. Since the longitude $\lambda_{A}$ of the Klein bottle
is oriented consistent with the saddle orbit $S$ and $H$ transforms the orbit
$S$ into the orbit $S^{\prime}$ with orientation preservation, then
$\lambda_{{A^{\prime}}}$ is oriented consistent with the saddle orbit
$S^{\prime}$. On the other side, by Lemma 2,

$\langle\lambda_{A}\rangle=\delta_{A}\langle\mathcal{G}_{A}\rangle,\,\langle\lambda_{A^{\prime}}\rangle=\delta_{A^{\prime}}\langle\mathcal{G}_{A^{\prime}}\rangle,$

whence, by virtue of a simple chain of equalities

$\delta_{A^{\prime}}\langle\mathcal{G}_{A^{\prime}}\rangle=h_{A*}(\delta_{A}\langle\mathcal{G}_{A}\rangle)=\delta_{A}\langle\mathcal{G}_{A^{\prime}}\rangle,$

we get that $\delta_{A}=\delta_{A^{\prime}}$. It is proved similarly that
$\delta_{R}=\delta_{R^{\prime}}$. Thus $C_{f^{t}}=C_{f^{\prime t}}$.

Sufficiency. Let the flows $f^{t}$ and $f^{\prime t}$ have equal invariants
$C_{f^{t}}=(\delta_{A},\delta_{R}),\ C_{f^{\prime
t}}=(\delta_{A^{\prime}},\delta_{R^{\prime}})$. Let us show that the flows
$f^{t}$ and $f^{\prime t}$ are topologically equivalent.

Proposition 1 implies that the homeomorphism

$H\big{|}_{V_{S}}=H_{S^{\prime}}H_{S}^{-1}\colon V_{S}\to V_{S^{\prime}}$

is topological equivalence homeomorphism of the flows $f^{t}\big{|}_{V_{S}}$
and $f^{\prime t}\big{|}_{V_{S^{\prime}}}$. It remains to extend this
homeomorphism to $\mathcal{V}_{A}$ and $\mathcal{V}_{R}$.

The homeomorphism $H$ is already defined on the set $\mathcal{T}_{A}\cap
T_{S}$, which is a tubular neighborhood $N(K_{A})$ of the Klein bottle
$K_{A}$. By item 2) of Proposition 3 the set $V(K_{A})=T_{A}\setminus{\rm
int}\,N(K_{A})$ is a solid torus whose meridian is homotopic to the meridian
$\mu_{A}$ of the Klein bottle $K_{A}$ to $N(K_{A})$. It follows from the
properties of the homeomorphism $H$ that $K_{A^{\prime}}=H(K_{A})$ and
$N(K_{A^{\prime}})=H(N(K_{A}))$ is a tubular neighborhood of the Klein bottle
$K_{A^{\prime}}$. By point 2) of Proposition 3 the set
$V(K_{A^{\prime}})=T_{A^{\prime}}\setminus{\rm int}\,N(K_{A^{\prime}})$ is a
solid torus whose meridian is homotopic to the meridian $\mu_{A^{\prime}}$ of
the Klein bottle $K_{A^{\prime}}$ in $N(K_{A^{\prime}})$. Since any
homeomorphism of the Klein bottle does not change the homotopy class of the
meridian (see, for example, [7, Lemma 5]), the homeomorphism $H\colon\partial
V(K_{A})\to\partial V(K_{A^{\prime}})$ extends to the homeomorphism $H\colon
V(K_{A})\to V(K_{A^{\prime}})$ (see, for example, [12, Exercise 2E5]). Thus
$H$ is defined on $\mathcal{T}_{A}$ and $\mathcal{T}_{R}$.

Since the parallel $\lambda_{A}\,(\lambda_{{A^{\prime}}})$ of the Klein bottle
is oriented consistent with the saddle orbit $S\,(S^{\prime})$ and $H$
transforms the orbit $S$ into the orbit $S^{\prime}$ orientation-preserving,
then $H_{*}(\langle\lambda_{A}\rangle)=\langle\lambda_{{A^{\prime}}}\rangle$.
Since $\delta_{A}=\delta_{A^{\prime}}$, then
$H_{*}(\delta_{A}\langle\lambda_{{A}}\rangle)=\delta_{A^{\prime}}\langle\lambda_{{A^{\prime}}}\rangle$
and hence
$H_{*}(\langle\mathcal{G}_{A}\rangle)=\langle\mathcal{G}_{A^{\prime}}\rangle$.
By Proposition 2 the homeomorphism $H\big{|}_{\mathcal{T}_{A}}$ extends to
$\mathcal{V}_{A}$ by a homeomorphism realizing the equivalence of flows
$f^{t}\big{|}_{\mathcal{V}_{A}}$ and $f^{\prime
t}\big{|}_{\mathcal{V}_{A^{\prime}}}$. Similarly, $H$ can be extended to
$\mathcal{V}_{R}$. Thus, the homeomorphism $H$ is defined on the whole $M^{4}$
and realizes the equivalence of the flows $f^{t},\,f^{\prime t}$. ∎

##  5 Realization of flows by admissible set

In this section, we will prove Theorem 2: for any element
$C\in\mathbb{S}^{0}\times\mathbb{S}^{0}$ there is a flow $f^{t}\in
G_{3}^{-1}(M^{4})$ such that $C=C_{f^{t}}$.

###### Proof.

Let us construct the flow
$f^{t}\colon\mathbb{S}^{2}\times\mathbb{S}^{1}\to\mathbb{S}^{2}\times\mathbb{S}^{1}$
with the invariant $C_{f^{t}}=(+1,+1)$ as a suspension over the sphere
diffeomorphism $\zeta\colon\mathbb{S}^{2}\to\mathbb{S}^{2}$ with three
periodic orbits. To do this, we describe the construction of the
diffeomorphism $\zeta$.

Let the function $\sigma\colon\mathbb{R}_{+}\to[0,1]$ be given by the formula
(see Fig. 2):

$\sigma(x)=\begin{cases}\exp{\left(2-\dfrac{1}{\left(\dfrac{1}{x}-2\right)^{2}}-\dfrac{1}{(x-2)^{2}}\right)},&x\in\left[\frac{1}{2},2\right]\\\
0,&x\notin\left[\frac{1}{2},2\right].\end{cases}$

Note that $\sigma(1/x)=\sigma(x).$

Figure 2: Graph of the function $\sigma(x)$.

Introduce spherical coordinates in $\mathbb{R}^{3}$ by
$x=r\cos\varphi\cos\theta,\ y=r\cos\varphi\sin\theta,z=r\sin\varphi,\quad
r>0,\ \varphi\in[-\pi;\pi),\ \theta\in[-\pi/2;\pi/2)$. Let
$\chi^{t}\colon\mathbb{R}^{3}\to\mathbb{R}^{3}$ be the flow defined by system
of equations:

$\begin{cases}\dot{r}=\begin{cases}-r(r-1),&0\leqslant r\leqslant 1\\\
1-r,&1\leqslant r\end{cases}\\\ \dot{\varphi}=-\sigma(r)\sin\varphi\\\
\dot{\theta}=-\sigma(r)\theta\end{cases}$

and the diffeomorphism $q\colon\mathbb{R}^{3}\to\mathbb{R}^{3}$ be defined by
the formula:

$q(x,y,z)=(1/x,\ -1/y,\ 1/z).$

Figure 3: Flow $\chi^{t}$ phase portrait

Using stereographic projection (see Figure 4)
$\vartheta\colon\mathbb{S}^{3}\setminus\\{N\\}\to\mathbb{R}^{3}$
($N=(0,0,0,1),S=(0,0,0,-1)$) by the given formula:

$\vartheta(x_{1},x_{2},x_{3},x_{4})=\left(\frac{x_{1}}{1-x_{4}},\frac{x_{2}}{1-x_{4}},\frac{x_{3}}{1-x_{4}}\right).$

Figure 4: Stereographic projection

project the diffeomorphism $q\chi^{1}$ onto $\mathbb{S}^{3}$:

$f(x)=\begin{cases}\vartheta^{-1}q\chi^{1}\vartheta(x),&x\notin\\{N,S\\}\\\
S,&x=N,\\\ N,&x=S\end{cases}$

Then the flow $f^{t}=[f]^{t}$ belongs to the class $G^{-1}_{3}(M^{3})$ and
$C_{f^{t}}=(+1,+1)$. By construction, $f$ is an orientation-preserving
diffeomorphism of the 3-sphere, and hence the ambient manifold of the
suspension $[f]^{t}$ is homeomorphic to $\mathbb{S}^{3}\times\mathbb{S}^{1}$.

We construct the rest of the flows of class $G^{-1}_{3}(M^{4})$ by modifying
the constructed flow $f^{t}$.

Let $\mathcal{O}$ be an attractive or repelling periodic orbit of $f^{t}$ and
$V_{\mathcal{O}}=V^{0}_{\mathcal{O}}$ be its canonical neighborhood,
$V^{t}_{\mathcal{O}}=f^{t}(V^{0}_{\mathcal{O}})$. Without loss of generality,
we assume that $V^{-1}_{A}\cap V^{1}_{R}=\emptyset$. Let $\vec{v}(x)$ denote
the vector field induced by the flow $f^{t}$ on
$\mathbb{S}^{3}\times\mathbb{S}^{1}$. For points $x$ that belong to the basin
of the orbit $\mathcal{O}$, let $\vec{n}_{{\mathcal{O}}}(x)$ denote the field
of unit outward normals to the hypersurfaces $\partial V^{t}_{\mathcal{O}}$
and let $s_{\mathcal{O}}(x)\in\mathbb{R}$ be the time such that
$f^{s_{\mathcal{O}}(x)}(x)\in V^{0}_{\mathcal{O}}$. We define the vector field
$\vec{v}^{\prime}(x)$ on $\mathbb{S}^{3}\times\mathbb{S}^{1}$ by the formulas

$\vec{v}^{\prime}(x)=\begin{cases}(1-s_{A}(x))(1+s_{A}(x))\vec{n}(x)+s_{A}(x)\vec{v}(x),&x\in
V_{A}^{-1}\setminus V^{1}_{A}\\\
(1-s_{R}(x))(1+s_{R}(x))\vec{n}(x)+s_{R}(x)\vec{v}(x),&x\in V_{R}^{1}\setminus
V^{-1}_{R}\\\ v(x),&\text{ otherwise}\end{cases}$

and denote by $f^{\prime t}$ the flow he induced on
$\mathbb{S}^{3}\times\mathbb{S}^{1}$. For $\delta\in\\{-1,+1\\}$ we define the
diffeomorphism
$w_{\delta}\colon\mathbb{S}^{3}\times\mathbb{S}^{1}\to\mathbb{S}^{3}\times\mathbb{S}^{1}$
by the formula

$w_{\delta}(x,y)=(x,\delta y).$

For $C=(\delta_{A},\delta_{R})\in\mathbb{S}^{0}\times\mathbb{S}^{0}$ we define
the flow $f^{t}_{C}$ on $\mathbb{S}^{3}\times\mathbb{S}^{1}$ by the formula

$f^{t}_{C}(x)=\begin{cases}w_{\delta_{A}}f^{\prime t}w_{\delta_{A}}(x),&x\in
V_{A}\\\ w_{\delta_{R}}f^{\prime t}w_{\delta_{R}}(x),&x\in V_{R}\\\
f^{t},&\text{ otherwise.}\end{cases}$

It is easy to see that the flow’s $f^{t}_{C}$ invariant is $C$ and its ambient
manifold is $\mathbb{S}^{3}\times\mathbb{S}^{1}$. ∎

##  6 Ambient manifolds of the flow of class $G^{-}_{3}(M^{4})$

In this section, we prove Theorem 3: the only 4-manifold admitting
$G^{-}_{3}(M^{4})$ flows is $\mathbb{S}^{3}\times\mathbb{S}^{1}$. Moreover,
the set $G^{-}_{3}(\mathbb{S}^{3}\times\mathbb{S}^{1})$ consists of exactly
eight equivalence classes of the considered flows.

###### Proof.

Assume that $f^{t}\in G^{-}_{3}(M^{4})$ then by Theorem 1 $f^{t}$ is
topologically equivalent to either the flow $f^{t}_{C}$ or flow $f^{-t}_{C}$
for some $C\in\mathbb{S}^{0}\times\mathbb{S}^{0}$. And since the topological
equivalence of flows implies the homeomorphism of their ambient manifolds, the
supporting manifold of the flow $f^{t}$ is homeomorphic to
$\mathbb{S}^{3}\times\mathbb{S}^{1}$.

Since the elements of the set $C\in\mathbb{S}^{0}\times\mathbb{S}^{0}$
correspond one-to-one to the equivalence classes of flows from
$G^{-1}_{3}(M^{4})$, the family $G_{3}^{-}(M^{4})=G^{-2}_{3}(M^{4})\sqcup
G^{-1}_{3}(M^{4})$ contains $8$ topological equivalence classes. ∎

## References

  * [1] Daniel Asimov.  Round handles and non-singular morse-smale flows.  Annals of Mathematics, 102(1):41–54, 1975.
  * [2] Christian Bonatti, Viatcheslav Grines, and Elisabeth Pecou.  Two-dimensional links and diffeomorphisms on 3-manifolds.  Ergodic Theory and Dynamical Systems, 22(3):687–710, 2002.
  * [3] Beatriz Campos, Alicia Cordero, Jose Martínez Alfaro, and P Vindel.  Nms flows on three-dimensional manifolds with one saddle periodic orbit.  Acta Mathematica Sinica, 20(1):47–56, 2004.
  * [4] Vyacheslav Grines, Timur Medvedev, and Olga Pochinka.  Dynamical systems on 2-and 3-manifolds, volume 46.  Springer, 2016.
  * [5] M.C. Irwin.  A classification of elementary cycles.  Topology, 9(1):35–47, 1970.
  * [6] Czes Kosniowski.  A first course in algebraic topology.  Cambridge University Press, 1980.
  * [7] WB Raymond Lickorish.  Homeomorphisms of non-orientable two-manifolds.  In Mathematical Proceedings of the Cambridge Philosophical Society, volume 59, pages 307–317. Cambridge University Press, 1963.
  * [8] Olga Pochinka and Danila Shubin.  On 4-dimensional flows with wildly embedded invariant manifolds of a periodic orbit.  Applied Mathematics and Nonlinear Sciences, 5(2):261–266, 2020\.
  * [9] Olga Pochinka and Danila Shubin.  Non-singular morse–smale flows on n-manifolds with attractor–repeller dynamics.  Nonlinearity, 35(3):1485, 2022.
  * [10] Olga Pochinka and Danila Shubin.  Nonsingular morse–smale flows with three periodic orbits on orientable $3$-manifolds.  Mathematical Notes, 112(3):436–450, 2022.
  * [11] Olga Pochinka and Danila Shubin.  Topology of ambient 3-manifolds of non-singular flows with twisted saddle orbit.  arXiv preprint arXiv:2212.13224, 2022.
  * [12] Dale Rolfsen.  Knots and links, volume 346.  American Mathematical Soc., 2003.
  * [13] Danila Shubin.  Topology of ambient manifolds of nonsingular flows with three twisted orbits (in russian).  Izvestiya Vysshikh uchebnykh zavedeniy. Prikladnaya nelineynaya dinamika, 29(6):863–868, 2021.
  * [14] Stephen Smale.  Differentiable dynamical systems.  Bulletin of the American mathematical Society, 73(6):747–817, 1967\.
  * [15] Ya. L. Umanskii.  Necessary and sufficient conditions for topological equivalence of three-dimensional morse–smale dynamical systems with a finite number of singular trajectories.  Matematicheskii Sbornik, 181(2):212–239, 1990.


[subsection]section *[subsubsection]subsection

On the Impact of Biological Risk in Aquaculture Valuation and Decision Making

Christian-Oliver Ewald111Department of Mathematics and Statistics, Umeå
University, Sweden,222Inland University of Applied Sciences, Lillehammer,
Norway,333E-mail<EMAIL_ADDRESS>and<EMAIL_ADDRESS>Kevin
Kamm11footnotemark: 1,33footnotemark: 3,

August 28, 2024

This paper explores the impact of stochastic mortality and disease on animal-
based commodities, with a specific emphasis on aquaculture, particularly in
the context of salmon farming. The investigation delves into the stochastic
nature of mortality and treatment plans based on historical data related to
salmon lice. Given that salmon lice pose a significant challenge in salmon
farming, with associated treatment costs estimated to be comparable to feeding
expenses, their removal is imperative to ensure the survival of the salmon and
comply with the Norwegian government’s stipulation of maintaining 0.5 lice per
fish.

We propose a new model that considers the relationship between hosts and
parasites to estimate the number of treatments required and their overall
cost. An important aspect of this model is its incorporation of stochastic
effectiveness for each removal. After calibrating the model to the available
data, the study examines the stochastic behavior’s impact on the optimal
harvesting decision in comparison to deterministic mortality models. The
results indicate an approximate $1.5\,\%$ increase in the value of the salmon
farm when employing the harvesting rule based on the stochastic host-parasite
model as opposed to a deterministic model.

eep Learning, Optimal Stopping, Real Options, Host-Parasite Model, Aquaculture

Code availability:  The Matlab/ Python code and data sets to produce the
numerical experiments are available at
https://github.com/kevinkamm/AquacultureStochasticMortality.

##  1 Introduction

This study draws parallels with research conducted by [4], which explored the
influence of stochastic feeding costs on optimal decision-making in
aquaculture management. However, we will focus in this article on the effects
of stochastic mortality instead of feeding. Our commodity models align with
those outlined in [4].

Specifically, we examine stochastic mortality models within the context of
salmon farms. We do not focus on the intrinsic Salmon mortality, but rather
the mortality due to parasites known as salmon lice. These parasites attach
themselves to the salmon’s skin and feed on its blood. If left unchecked,
infestation can lead to mortality, which can harm neighboring fish farms. To
prevent adverse effects, the Norwegian government has set a threshold of 0.5
female lice per fish. As a result, farmers must take countermeasures to stay
below this threshold.

Salmon lice treatments can be classified into three main categories: medical,
biological, and mechanical. It is worth noting that biological treatments have
been phased out due to ethical concerns surrounding the sacrifice of other
fish species. Currently, the predominant method involves mechanical removal,
which includes techniques such as warm or freshwater baths, brushing, or laser
methods. Even though very small, these methods pose some risks to salmon as
well, which are carefully considered in this study.

Accurate modeling of treatment times and associated costs is crucial to
optimize the value of a fish farm and determine the optimal harvesting time.
According to [10], biological costs are now as significant as feeding costs,
highlighting the need for a comprehensive examination of stochastic mortality.

Our contribution involves introducing a novel model for treatment times and
their impact on salmon mortality, rooted in the biological dynamics of the
problem. Drawing inspiration from [1], we employ a host-parasite model. We
demonstrate the calibration of this model to data and assess the impact of
stochasticity on its deterministic counterpart, represented by the mean of the
model. The results suggest a $1.5\,\%$ increase in farm value when using a
stochastic model.

##### Literature review.

For an overview of the historical development of salmon commodity pricing and
valuation models, we refer the reader to [5]. Additionally, for a
comprehensive treatment of all the economic factors of fish farming, we refer
the reader to [10] and [8]. For a treatment of stochastic feeding costs, we
refer the reader to [4]. In this review, we focus on (stochastic) mortality.

Deterministic host-parasite models have been extensively explored in
biological contexts, with notable studies conducted by [1], [9], [12], among
others. In particular, [6] considered a deterministic model specifically for
salmon lice, which is related to the model considered in this paper. In
addition, [7] have examined lice thresholds for both individual farms and farm
networks to determine the level of lice per fish that warrants treatment. They
used a deterministic model of parasite growth with a simplified harvesting
rule. In addition, [2] and [11] explored sophisticated models that describe in
detail the spread of lice taking hydrodynamic effects into account.

To the best of our knowledge, this article is the first to study harvesting
rules that depend on stochastic mortality caused by salmon lice.

The remainder of this paper is structured as follows: In Section 2, we
introduce the commodity models, followed in Section 2.1 by a description of
the salmon farm features considered in this paper. The mortality model will be
introduced in Section 2.1.1. Afterwards, in Section 2.2, we define our optimal
stopping problem using both deterministic and stochastic feeding costs. The
numerical results will be discussed in Section 3, which first explains the
market data in Section 3.1 and salmon lice data in Section 3.2, followed by a
calibration algorithm in Section 3.3 for the model. Then, we explain our
methodology for comparing stochastic and deterministic feeding rules in
Section 3.4 and reach our conclusion in Section 4.

##  2 Mathematical and Model Framework

Following [4], let henceforth $\left(\Omega,\mathcal{F},\mathbb{Q}\right)$ be
a probability space and $\mathbb{Q}$ be a risk-neutral measure. Moreover, let
$T>0$ be a finite time horizon and $r>0$ a fixed deterministic interest rate.
We will use a multi-commodity framework consisting of two independent
Schwartz-2-factor models directly under $\mathbb{Q}$, i.e.

$\displaystyle dS_{t}^{i}$
$\displaystyle=\left(r-\delta_{t}^{i}\right)S_{t}^{i}dt+\sigma^{i}_{1}S_{t}^{i}dW_{t}^{i,1},\quad
S_{0}^{i}=s_{0}^{i}\in\mathbb{R}_{\geq 0,}$ $\displaystyle d\delta_{t}^{i}$
$\displaystyle=\left(\kappa^{i}\left(\alpha^{i}-\delta^{i}_{t}\right)-\lambda^{i}\right)dt+\sigma_{2}^{i}dW_{t}^{i,2},\quad\delta_{0}^{i}=d_{0}^{i}\in\mathbb{R}$
$\displaystyle d\langle W^{i,1},W^{i,2}\rangle_{t}$
$\displaystyle=\rho^{i}dt,\qquad d\langle W^{i,1},W^{j,2}\rangle_{t}=0,\quad
i\neq j,$

where $W_{t}\coloneqq\left(W_{t}^{i,j}\right)_{i,j=1,2}$ are Brownian motions
generating the $\sigma$-algebra $\mathcal{F}_{t}$ augmented by $\mathbb{Q}$
nullsets satisfying the usual conditions.

The dynamics $dS_{t}^{i}$ describe the i-th commodity’s spot price with
convenience yield described by $d\delta_{t}^{i}$. The parameter
$\sigma_{1}^{i}>0$ is the spot volatility, $\kappa^{i}>0$ the mean reversion
speed of the convenience yield, $\alpha^{i}\in\mathbb{R}$ the long-term mean,
$\lambda^{i}\geq 0$ a risk-premium and $\sigma_{2}^{i}>0$ the volatility of
the convenience yield. For a treatment of future prices in this framework we
refer the reader to [4] since this will not be the main focus of this article.

###  2.1 Salmon Farm Parameters

In this subsection, we briefly describe the features of a salmon farm, which
we consider in this paper, and refer the reader to [10] for a detailed
economical treatment of aquaculture farms.

We will assume a single rotation fish farm, which means that we assume that a
fish farmer buys $\mathrm{H}_{0}\in\mathbb{N}$ smolt, young salmon, and feeds
them till they are ready for harvest, and after the harvest, our consideration
is finished. The growth of the fish over time depends on various factors like
water temperature, amount and quality of feed, health, etc. We simplify this
by considering a deterministic function over time called _Bertalanffy’s growth
function_ , which is given by

$\displaystyle\mathrm{w}_{t}$
$\displaystyle\coloneqq\mathrm{w}_{\infty}\left(a-b\,e^{-ct}\right)^{3},$

for some parameters $a,b,c$ given in Table 1. This function measures the
growth in kg per fish. Hence, we need to keep track of our initial population
$n_{0}$ over time as well.

####  2.1.1 Stochastic Mortality and Treatment Costs

In our previous article [4], we assumed a fixed rate of mortality
$\mathrm{m}>0$. The total number of fishes $\mathrm{H}_{t}$ with mortality
rate $\mathrm{m}$ was modeled as another deterministic function

$\displaystyle d\mathrm{H}_{t}$
$\displaystyle=-\mathrm{m}\,\mathrm{H}_{t}dt,\quad\mathrm{H}_{0}>0.$

Then, we can measure the total biomass (in kg) of the fish farm over time by
setting

$\displaystyle\mathrm{B}_{t}=\mathrm{H}_{t}\mathrm{w}_{t}.$

Now, we want to extend this model by using a more sophisticated model for
$\mathrm{H}_{t}$ and meanwhile model the number of mechanical lice removals,
such that the threshold of $l=0.5$ female lice per fish is not exceeded. It is
straightforward to include time-varying thresholds $l$ and indeed they
sometimes vary between $0.2$ and $0.5$ depending on the region and time of the
year.

##### Host-parasite model.

Let us denote by $H_{t}$ the host population and $P_{t}$ the parasite
population. A notable difference to standard SIR models (Susceptible-
Infectious-Recovered) is that we can allow an infected species to be infected
again, i.e., have more than one louse. Therefore, we need to model the number
of lice and fish as a system of coupled dynamics. Since the lice threshold
shall not be exceeded, otherwise extermination of the farm would occur under
some circumstances, we will include the threshold explicitly into the
dynamics. We only model mechanical removals. They have a large success rate of
removals but are usually only performed on a part of the farm instead of the
whole site. Thus, many removals are usually needed during the growing period
and each removal introduces some cost. During each removal, we will assume a
random success rate of lice removal and a small random number of salmon not
surviving the treatment, e.g., hot water baths can be stressful for the salmon
as well, mechanical brushes can hurt the salmon, etc.

Let $0\leq t_{1}\leq t_{2}\leq\dots$ be the times of the treatments, i.e. the
times such that $\frac{P_{t}}{H_{t}}\geq l_{t}$, then we define our host-
parasite model for $t<t_{1}$ as for generic processes $h_{t}$ and $p_{t}$
determined below

$\displaystyle dH_{t}$ $\displaystyle=h_{t}(P_{t},H_{t})dt,\quad H_{0}>0$
$\displaystyle dP_{t}$ $\displaystyle=p_{t}(P_{t},H_{t})dt,\quad P_{0}>0$

then restart at for $t\in[t_{1},t_{2})$

$\displaystyle dH_{t}$ $\displaystyle=h_{t}(P_{t},H_{t})dt,\quad
H_{t_{1}}=X_{t_{1}}\,H_{t_{1}-}$ $\displaystyle dP_{t}$
$\displaystyle=p_{t}(P_{t},H_{t})dt,\quad P_{t_{1}}=Y_{t_{1}}\,P_{t_{1}-}$

and so on.

Since we are no experts in biology, especially salmon lice, we restrict
ourselves to a simple model not taking the different stages of parasites in
their life-cycle into account. We encourage our readers to modify the model
according to their requirements and test it using the publicly available code.
To keep our experiments simple, we have used deterministic dynamics for the
lice, which can be estimated from biological experiments. Although the growth
of parasites and salmon depends on various factors such as water temperature,
salinity, and other random factors, we have assumed that the farmer has a good
understanding of these parameters for their farm, and hence, we have excluded
them from our dynamics. In particular, we chose exponential growth models
based on [1]

$\displaystyle\begin{aligned}
h_{t}(P_{t},H_{t})&=-\left(\mu+\alpha\frac{P_{t}}{H_{t}}\right)H_{t},&&X_{t}\sim\mathrm{unif}(0.995,1)\\\
p_{t}(P_{t},H_{t})&=\left(\lambda\frac{H_{t}}{H_{0}}-(b+\mu)-\alpha\frac{P_{t}}{H_{t}}\right)P_{t},&&Y_{t}\coloneqq
0.1+0.8\tilde{Y}_{t},\
\tilde{Y}_{t}\sim\mathrm{beta}(\beta_{1},\beta_{2})\end{aligned}$ (1)

We chose $Y_{t}$ in this way to ensure a minimal success rate of $10\,\%$ and
a maximal success rate of $90\,\%$, which sounded more realistic and avoided
pathological examples with zero and complete success rates. Moreover, we will
denote by $N_{t}$ the counting process for the necessary mechanical removals
per path, i.e., the hitting times $\frac{P_{t}}{H_{t}}=l(t)$, which is well
defined, since the paths of $H_{t},P_{t}$ prior to the hitting boundary are
continuous.

The parameter $\mu$ controls the intrinsic mortality of salmon and will be
fixed to $5\,\%$ for our tests and similarly $b$ controls the intrinsic
mortality of salmon lice and will also be fixed to $5\,\%$. The salmon
mortality due to salmon lice can be adjusted by $\alpha$ and increases
multiplicatively with the ratio of lice per fish. We will fix it to $10\,\%$
in our experiments, which might seem high but we will see that the impact of
this factor is small as long as the lice per fish is kept small as well, which
is the purpose of the treatments. The salmon population cannot grow over time,
so there are no births, but the lice population will increase exponentially if
left unchecked. This rate of population growth $\lambda$ depends in this model
on the relative number of salmon in the farm and will be calibrated from the
data alongside the effectiveness of lice removals by finding suitable
parameters $\beta_{1}$ and $\beta_{2}$ for the beta-distribution.

We tried other distributions as well, but the beta distribution matched the
data best. Also, the mortality due to treatments is very sensitive to the
lower bound of the uniform random number. Changing the lower bound to $0.99$
leads to an overestimation of mortality.

##### Treatment costs.

We will assume for simplicity that each treatment will remove a certain
percentage from the current value of the salmon farm, i.e. the total treatment
costs are given by

$\displaystyle\mathrm{Tc}_{t}=\mathrm{tc}\sum_{0\leq s\leq t}{\Delta
H_{s}},\quad\Delta H_{s}\coloneqq H_{s}-H_{s-},\quad\mathrm{tc}\in[0,1].$

The treatment times are identified by the jumps in the host population process
$\Delta H_{s}$ and are in the host-parasite model the times, where the lice-
threshold $l$ is exceeded.

We want to compare the impact of the stochasticity to the harvesting decision
rule. For this, we want to compare the harvesting decisions using stochastic
and deterministic mortality. Similar to [4], to make these models comparable,
we derive the deterministic models by taking the mean of the stochastic model,
i.e.

$\displaystyle
H_{t}^{\text{determ}}\coloneqq\mathbb{E}\left[H_{t}\right],\quad\mathrm{Tc}_{t}^{\text{determ}}\coloneqq\mathbb{E}\left[\mathrm{Tc}_{t}\right].$

This also implies that the biomass
$\mathrm{B}_{t}=\mathrm{H}_{t}\mathrm{w}_{t}$ in this model will either be
stochastic or deterministic depending on the choice of $\mathrm{H}_{t}$.

####  2.1.2 Feeding and Harvesting Costs

We will consider only two varying factors of production costs: harvesting
costs and feeding costs in this article, all the other costs like labor,
medical treatments, capital costs, etc., will be treated as constants for
simplicity. The harvesting costs are given by $\mathrm{h}_{0}$ per kg of fish
and the total harvesting cost of the fish farm at time $t>0$ is therefore

$\displaystyle\mathrm{CH}_{t}=\mathrm{h}_{0}\,\mathrm{B}_{t}.$

For this, we set a conversion rate of how much kg of feed will convert to kg
of fish to

$\displaystyle\mathrm{c}\coloneqq 1.1\frac{\text{\,kg feed}}{\text{\,kg
fish}}.$

Let $F(0)=F_{0}$ be a given feeding cost for one fish per year. We will infer
the feeding costs from the relative changes of soybean prices $S_{t}^{2}$,
i.e.

$\displaystyle\tilde{S}_{t}^{2}\coloneqq\frac{S_{t}^{2}}{S_{0}^{2}},$

by using

$\displaystyle\mathrm{F}^{\text{stoch}}_{t}=\mathrm{F}_{0}\tilde{S}_{t}^{2},\quad\mathrm{F}^{\text{determ}}_{t}=\mathrm{F}_{0}\mathbb{E}\left[\tilde{S}_{t}^{2}\right],$

and define the discounted cumulative total feeding costs as

$\displaystyle\mathrm{CF}_{t}=\int_{0}^{t}{e^{-rs}\left(\mathrm{F}_{s}\,\mathrm{H}_{s}\left(\partial_{t}\mathrm{w}\right)(s)\,\mathrm{c}\right)ds}$

for general feeding costs $\mathrm{F}_{t}$.

The parameters in Table 1 are mostly taken from [5] and references therein.
The initial feeding, harvesting, and biological costs are estimated from [10,
p. 25 Figure 9 ]. We will use the values in Table 1 for the remainder of this
paper.

Table 1: Fish farm parameters. Name | Symbol | Value | Unit
---|---|---|---
Bertalanffy growth factor | $a$ | 1.113 | -
Bertalanffy growth factor | $b$ | 1.097 | -
Bertalanffy growth factor | $c$ | 1.43 | -
Asymptotic weight | $\mathrm{w}_{\infty}$ | 6 | kg
Mortality rate | $\mathrm{m}$ | 5 | %
Conversion rate | $\mathrm{c}$ | 1.1 | kg feed/ kg fish
Number of recruits | $\mathrm{H}_{0}$ | 10000 | fish
Time horizon | $T$ | 3 | years
Harvesting possibilities | $N$ | 72 | -
Salmon spot price | $\hat{S}_{0}^{1}$ | see (3) | NOK/ kg
Production costs | $\mathrm{PC}$ | $0.5\,\hat{S}_{0}^{1}$ | NOK/ kg
Harvesting costs | $\mathrm{h}_{0}$ | $0.1\,\mathrm{PC}$ | NOK/ kg
Feeding costs | $\mathrm{F}_{0}$ | $0.25\,\mathrm{PC}$ | NOK/ kg year
Biological costs | $\mathrm{BC}_{0}$ | $0.3\,\mathrm{PC}$ | -
Salmon initial value | $S_{0}^{1}$ | $\hat{S}_{0}^{1}-\mathrm{PC}+\mathrm{h}_{0}+\mathrm{F}_{0}+\mathrm{BC}_{0}$ | NOK/ kg
Soy initial value | $S_{0}^{2}$ | 1 | -
Treatment costs | $\mathrm{tc}$ | 1.5 | %

###  2.2 Optimal Stopping Problem

Following [4] the objective of a fish farmer is to find the optimal harvesting
time of the salmon cultivated in the farm, where optimal has to be understood
as obtaining the maximal expected value under a risk-neutral measure. This can
be formalized by considering the current value of the fish at their current
weight $S_{t}^{1}\mathrm{B}(t)$ minus the harvesting costs at $\mathrm{CH}(t)$
and the cumulative feeding costs up to this point in time $\mathrm{CF}(t)$. In
the case of deterministic feeding costs and the host-parasite model, let
$X_{t}^{\text{stoch}}\coloneqq\left(S_{t}^{1},\delta_{t}^{1},H_{t},P_{t}\right)$
and $X_{t}^{\text{determ}}\coloneqq\left(S_{t}^{1},\delta_{t}^{1}\right)$. In
the case of stochastic feeding costs $\left(S_{t}^{2},\delta_{t}^{2}\right)$
needs to be added. Thus, the dimension of $X_{t}$ is $d=6$ in the stochastic
host-parasite model with stochastic feeding costs and less for all the other
cases.

For a fixed choice of feeding model, the optimal stopping problem for
stochastic and deterministic mortality becomes respectively

$\displaystyle W_{0}^{\text{stoch}}(x)$
$\displaystyle\coloneqq\sup_{\tau^{\text{stoch}}}\mathbb{E}^{\mathbb{Q}}\left[\left.\exp\left(-r\tau\right)\left(\left(1-\mathrm{Tc}_{t}^{\text{stoch}}\right)S_{\tau}^{1}\mathrm{B}_{\tau}-\mathrm{CH}_{\tau}\right)-\mathrm{CF}_{\tau}\right|X_{0}^{\text{stoch}}=x\right],$
$\displaystyle W_{0}^{\text{determ}}(x)$
$\displaystyle\coloneqq\sup_{\tau^{\text{determ}}}\mathbb{E}^{\mathbb{Q}}\left[\left.\exp\left(-r\tau\right)\left(\left(1-\mathrm{Tc}_{t}^{\text{determ}}\right)S_{\tau}^{1}\mathrm{B}_{\tau}-\mathrm{CH}_{\tau}\right)-\mathrm{CF}_{\tau}\right|X_{0}^{\text{determ}}=x\right].$

In this paper, we compare the stopping rule obtained from
$W_{0}^{\text{stoch}}(x)$ and $W_{0}^{\text{determ}}(x)$ by evaluating

$\displaystyle
V_{0}^{z}(x)\coloneqq\mathbb{E}^{\mathbb{Q}}\left[\left.\exp\left(-r\tau^{z}\right)\left(\left(1-\mathrm{Tc}_{t}^{\text{stoch}}\right)S_{\tau^{z}}^{1}\mathrm{B}_{\tau^{z}}-\mathrm{CH}_{\tau^{z}}\right)-\mathrm{CF}_{\tau^{z}}\right|X_{0}^{\text{stoch}}=x\right]$
(2)

for $z=\text{stoch},\,\text{determ}$ in an appropriate way described in
Section 3.4, and answer the question whether $V_{0}^{\text{determ}}(x)$ is a
good approximation of $W_{0}^{\text{stoch}}(x)$ or if it is beneficial to
consider the slightly more complicated stopping rule $\tau^{\text{stoch}}$.

We will solve the optimal stopping problems numerically by using the Deep
Optimal Stopping Network by [3] and refer for the algorithmic details to the
same paper. A simple least-square Monte Carlo (LSMC) approach with a
polynomial basis, led to numerically ill-conditioned matrices for solving the
regression step and inaccurate results. We leave it to future research to find
a more suitable basis, e.g., a neural network.

##  3 Numerical Experiments

In this section, we will perform our numerical experiments. In Section 3.1, we
will briefly discuss the historical data used for the commodity models, as
well as fix their parameters. This is followed by a detailed examination of
the available salmon lice data in Section 3.2, which will be used to calibrate
the host-parasite model in Section 3.3. After the calibration, we will compare
the stochastic mortality model to its deterministic counterpart in Section 3.4
and reach our conclusion that it can be beneficial to consider stochastic
mortality models.

For the calibration we used Matlab 2023a with the (Global) Optimization
Toolbox and for the DNN (Intel-)Python 3.10 with Tensorflow 2.8.0 running on
Windows 11 Pro, on a machine with the following specifications: processor
Intel(R) Core(TM) i9-13900K CPU @ 3.00 GHz and 2x32 GB (Dual Channel) Kingston
DIMM DDR5 RAM @ 5600 MHz. A GPU did significantly improve the performance in
this case and we used an NVIDIA GeForce RTX 4090 (24 GB GDDR6X RAM).

For all of our experiments, we will fix the interest rate to $r=0.0303$.

###  3.1 Future Market Data

We use the same historical future data 01/01/2012 till 01/06/2023 for both
salmon and soy as in [4] and chose the following parameters that had been
obtained there:

$\displaystyle\begin{aligned} \sigma_{1}^{1}=0.23,\ \sigma_{2}^{1}=0.75,\
\kappa^{1}=2.6,\ \alpha^{1}=0.02,\ \lambda^{1}=0.2,\ \rho^{1}=0.9,\
\delta^{1}_{0}=0.57,\ \hat{S}_{0}^{1}=95,\\\ \sigma_{1}^{2}=1,\
\sigma_{2}^{2}=0.4,\ \kappa^{2}=1.2,\ \alpha^{2}=0.06,\ \lambda^{2}=0.14,\
\rho^{2}=0.44,\ \delta^{2}_{0}=0,\ \hat{S}_{0}^{2}=1500.\end{aligned}$ (3)

###  3.2 Lice and Treatment Data

In this section, we explain what data is available for estimating stochastic
mortality and treatment costs.111The data we used is publicly available at
https://www.barentswatch.no/nedlasting/fishhealth/lice (last accessed
31/07/2023 13:04 CEST). The data contains information about salmon lice per
fish, moving lice per fish, and stuck lice per fish, as well as removal and
treatment measures to counteract the lice infestation for each fishing site in
Norway. These counts are updated weekly since 01/01/2012 and are available
till today.

We are aware, that the lice growth is affected by the salinity and temperature
of the water and therefore restrict ourselves to farms in a specific region of
Norway, namely Trøndelag, to avoid dealing with immense fluctuations of those.

There are three major categories of lice removals: Cleaner fish, medical
treatments, and mechanical removals. In Figure 1, we can see the chosen
treatments and treatment times depicted by colored marks of roughly 200 fish
farms. The usage of cleaner fish (red dots) was very popular from 2012 till
roughly 2018 but is not used anymore in more recent years. Also, the usage of
medical treatments (yellow dots) seems to have lessened over time. On the
other hand, mechanical removals (blue dots) of lice have become the method of
choice in more recent years and will be the focus of lice countermeasures in
this paper. To give an idea of what mechanical removal refers to, there are
methods using brushes to brush the lice of the salmon skin, hot-water or
fresh-water baths, as well as laser techniques. We will not distinguish
between different methods and note that each of these methods has a very small
probability of harming or killing the salmon in the process with varying
effectiveness of removing the lice.

Figure 1: Evolution of treatment techniques in Trøndelag from 2012 till today.

In Figure 2, we selected a random fish farm in Trøndelag region, to show the
evolution of female lice per fish (blue bold line), moving lice per fish (red
bold line), stuck lice per fish (yellow bold line), as well as the times of
using mechanical removals, medical treatments, or cleaner fish depicted with
blue, red, and yellow crosses respectively. The light-gray areas show the farm
periods from planting smolt (young salmon) to the harvesting of the farm. The
bold black line marks the maximal female lice per fish threshold, which must
not be exceeded for a longer period by Norwegian regulations of fish farms. If
a farmer would ignore this threshold or would use insufficient
countermeasures, the farm would be shut down and all fish would need to be
exterminated to prevent a salmon lice epidemic in the area.

We can see that the lice per fish can behave very differently in these periods
if we compare the earlier periods with the later ones, which may or may not be
due to the different treatments. Thus, for our investigation, we restrict
ourselves to datasets only having mechanical removals and no other treatments,
like in the last two periods. The green areas mark the sections of the periods
till the first mechanical removal. We will use these green sections to
estimate the reproduction rate of salmon lice in the host-parasite model.

Figure 2: Evolution of lice per fish, as well as countermeasures for one
specific salmon farm from 2012 till today.

In Figure 3, we show the roughly 100 selected green areas of all of the fish
farms in Trøndelag region. Each trajectory (in various colors) is one
evolution of female salmon lice per fish from the start of the fish farm till
a varying treatment time. There is no immediate clear pattern recognizable,
except for an exponential growth with varying rates. This can be due to
various factors, for example, preventive removals, i.e., not waiting till the
threshold is exceeded or close to it, a different number of lice per fish (in
different stages of their life cycles) at the start of the farm, different
rate of lice coming from the ocean, human error, etc. Since we are no experts
in this field, we will treat this as noise.

Figure 3: Evolution of lice per fish till first mechanical removal for farms
in Trøndelag, excluding all periods with additional treatments to mechanical
removals.

Now, that we have a better understanding of the evolution of lice, we focus in
Figure 4 on the distribution of accumulated mechanical removals from the start
till harvesting the fish farms. On the x-axis, we see the time in weeks,
starting in the back at zero and going up to 100 weeks in the front of the
picture. The y-axis shows the number of accumulated mechanical removals over
time and the z-axis the corresponding probability of this number. Hence, each
line (in the y-axis) for a fixed week is a probability distribution of the
accumulated number of removals over time. We see that in the first 10 weeks,
there is no treatment at all, then the entire mass is focused on one treatment
at roughly week 15, which spreads out afterwards. From week 70 onward (the
yellowish area), the distribution is quite stable and is because harvesting
will occur for the majority of the farms in this period, making further
treatments close to the harvesting time unnecessary. The mean of treatments in
the yellow area is roughly 10. We assumed that the total number of treatment
costs is roughly 30 % (see Table 1) of the production costs, which is 50 % of
the salmon value, and therefore 15 % of the overall value. This means that
$\mathrm{tc}=1.5\,\%$ is a good approximation for each treatment.

Figure 4: Evolution of distributions of cumulative mechanical removals for
farms in Trøndelag, excluding all periods with additional treatments to
mechanical removals.

###  3.3 Calibration of the Host-Parasite Model

In this section, we will explain how to calibrate the host-parasite model
introduced in Section 2.1.1 to our selection of the data, i.e., only farming
periods in Trøndelag region, where only mechanical removals were used as a
lice treatment leading to roughly 100 different farming periods.

We partition the calibration of the host-parasite model (1) into two steps:
First, we fit the deterministic part to the data shown in Figure 3, i.e., the
data of female lice per fish till the first mechanical removal, to infer the
reproduction rate of the parasites. Note that we will disregard for simplicity
the different phases that the copepodites undergo through their life cycle and
thus also the data on stuck lice per fish and moving lice per fish. After
inferring the reproduction rate of the salmon lice, we find the parameters for
the effectiveness $Y_{t}\sim\mathrm{beta}(\beta_{1},\beta_{2})$ to match the
distribution of cumulative mechanical removals in the yellow area of Figure 4.

Henceforth, let us fix the intrinsic mortality of salmon to $\mu=0.05$, the
mortality due to salmon lice infestation $\alpha=0.1$, and the intrinsic
mortality of salmon lice $b=0.05$. We found these parameters reasonable
without any biological validation at this point and encourage the reader to
change these values to their preferences in the code. As aforementioned, we
will infer in the first step the reproduction rate $\lambda$ from the data.
Thus, we want to solve the following least-squares problem

$\displaystyle\min_{\lambda>0}\sum_{k=1}^{K}\sum_{i=1}^{N}\left(\frac{P_{t_{i}}}{H_{t_{i}}}-\mathrm{LPF}^{k}_{t_{i}}\right)^{2},$

where $\mathrm{LPF}^{k}_{t_{i}}$ are the female lice per fish of the $k$-th
farm (y-axis in Figure 3) at time $t_{i}$ (x-axis in Figure 3). We solve the
coupled ODE for $H_{t}$ and $P_{t}$ by an explicit Euler-scheme with constant
step size $\Delta t=\frac{T}{10N-1}$ ($T$ and $N$ as in Table 1). Notice, that
$H_{t}$ and $P_{t}$ are deterministic till the first passing time
$\frac{P_{t}}{H_{t}}>l(t)$.

We found for the initial number of salmon $H_{0}=10000$ and $0.1\,\%$ lice per
fish the reproduction rate $\lambda=7.0143$.

In a second step, we simulate $M=1000$ trajectories of the host-parasite model
with jumps at the hitting times $\frac{P_{t}}{H_{t}}=l(t)$ with parameters
$\beta_{1},\beta_{2}$ for the effectiveness of the treatments. Hence, we want
to match the first two moments at $t=1.77$ (yellow area in Figure 4), i.e.

$\displaystyle\min_{\beta_{1}>0,\beta_{2}>0}\left(\mathbb{E}\left[\sum_{0\leq
s\leq t}{\Delta
H_{s}}\right]-\mathrm{Mean}\left[\mathrm{R}_{t}\right]\right)^{2}+\zeta\left(\mathrm{std}\left[\sum_{0\leq
s\leq t}{\Delta
H_{s}}\right]-\mathrm{std}\left[\mathrm{R}_{t}\right]\right)^{2},$

where $\zeta>0$ is a parameter to adjust the emphasis on the standard
deviation during the calibration222In our experiments, we found that $\zeta=2$
works best and use this choice in all of the following tests. and
$\mathrm{R}_{t}$ denotes the vector of cumulative mechanical removals up to
time $t$. We found the parameters $\beta_{1}=0.0829$ and $\beta_{2}=0.0281$.

We used Matlab 2023a’s lsqnonlin with the trust-region-reflective algorithm
for the first optimization problem, which took 0.1 seconds, and fmincon with
the interior point method for the second one, taking 19.8 seconds on CPU.

Figure 5: One trajectory of the simulated host model with treatments (bold red
line) and corresponding lice per fish in blue dots.

In Figure 5, we show one trajectory of the simulated host (red bold line) and
parasite population (dotted blue lines), as well as what would happen to the
salmon (red dashed line), if no lice-removals were applied in our model. The
left y-axis corresponding to the blue dots shows the lice per fish and the
right y-axis corresponds to the number of salmon. The x-axis shows the time-
axis in years. Let us focus on the red lines first. We can see that till
roughly $t=0.75$ they are the same because the model is deterministic till the
first hitting time. After $t=0.75$ the dotted line shows what would happen if
the lice were not removed, the salmon population would quickly decline in this
model. We also show two benchmarks in green and black dashed lines with
deterministic mortality rates of $\mathrm{m}=0.05$ and $\mathrm{m}=0.1$,
respectively. By the distance of the green and red dotted line, we can see the
impact of the increasing lice population. In this model, we can see that as
long as the lice per fish is below the threshold $0.5$, the mortality due to
lice has no significant impact on the salmon, which should reflect the reality
well. Now, let us focus on the lice per fish, i.e., the blue dots. We can see
the jump times well by the sudden drops, followed by an exponential increase
afterwards till the lice threshold is exceeded. The height of the jumps is
determined by the random numbers $Y_{t}$ drawn from the calibrated beta
distribution. At each of these jumps a uniform random number $X_{t}$ is drawn
as well to reflect the small probability of salmon dying from lice removal
treatments, which explains the jumps in the bold red line.

In Figure 5, we show histograms illustrating the distribution of cumulative
mechanical removals at two specific points in time $t=1.09$ on the left-hand
side and $t=1.77$ on the right-hand side. The blue bars correspond to the
simulated host-parasite model and the red bars to the data (slices of Figure
4).

Figure 6: Distribution of the number of cumulative mechanical removals using
the host-parasite model (blue bars). The left figure compares the model
distribution to the distribution from the data at $t=1.09$ and the right
figure at $t=1.77$.

We can see that the raw data exhibits two modes, while our model has a skewed
distribution with one mode. At $t=1.77$ they match well enough in our opinion,
since we can expect a lot of noise in this small dataset. For $t=1.09$ a
smaller number of mechanical removals is estimated by our model than seen from
the data. Including more points in time for the second calibration problem
could average these two fits, but we found the model distributions reasonable
for our further investigations.

###  3.4 Comparison of Exercise Decisions

In [4], we found that the difference in exercise decisions can be estimated
from a pathwise comparison of the stopping times. It led to the same
conclusions as determining the decision boundary first, so we will just use
the pathwise comparison here, which we will recall for the reader’s
convenience:

Let $\left(\Omega,\mathcal{F},\mathcal{F}_{t},\mathbb{P}\right)$ be a filtered
probability space satisfying the usual conditions, where
$\mathcal{F}_{t}\coloneqq\sigma\left(X_{s}|s\leq t\right)$ is generated by a
stochastic process, and $\mathcal{F}^{i}_{t}\subseteq\mathcal{F}_{t}$,
$i=1,2$, be two sub-filtrations. Moreover, let $\tau^{1}$ be an
($\mathcal{F}^{1}_{t}$-)stopping time and $\tau^{2}$ an
($\mathcal{F}^{2}_{t}$-)stopping time. Now, let us fix a path
$(X_{t}(\omega))_{t\geq 0}$, then conditioned on this path the two stopping
times $\tau^{1}(\omega)\in[0,T]$ and $\tau^{2}(\omega)\in[0,T]$ can be
compared as real-numbers.

Additionally, still conditioned on $(X_{t}(\omega))_{t\geq 0}$, we may compare
the stopped values of a stochastic process $Y_{t}$, i.e.,
$Y_{\tau^{1}(\omega)}$ and $Y_{\tau^{2}(\omega)}$.

We will compare now the impact of the stochastic mortality by comparing
$V_{0}^{\text{stoch}}$ and $V_{0}^{\text{determ}}$ using the stopping times
obtained from $W_{0}^{\text{stoch}}$ and $W_{0}^{\text{determ}}$ in the host-
parasite model.

As aforementioned, we solve the optimal stopping problem by using the DeepOS
network by [3], since in the standard LSMC approach the polynomial basis led
to poorly conditioned matrices for solving the linear systems, a ridged
regression did not help in this case. Using neural networks as basis functions
might improve the performance and we leave this for future considerations. The
training of the Deep Neural Networks with $1500+d$ epochs and a batch-size of
$2^{12}$ is quite fast using a GPU, it takes in the fully stochastic case
roughly 90 seconds using the host-parasite model and is a bit faster in the
other cases. The majority of the computational time comes from the simulation
of the stochastic processes. For the evaluation, we use $20\cdot 2^{12}$
trajectories and the evaluation of the neural networks takes less than a
second.

In Table 2, we show the relative improvement of using a stochastic mortality
model compared to deterministic mortality, denoted by
$\mathrm{RI}\coloneqq\frac{V_{0}^{\tau,\text{stoch}}}{V_{0}^{\tau,\text{determ}}}$.
Let us focus on the scenario with stochastic feeding costs first, i.e. the
second column. We can see that the mean stopping times are quite similar and
the relative improvement is roughly $1.5\,\%$.

Table 2: Relative improvements using a stochastic mortality model with deterministic and stochastic feeding costs. | Stochastic feeding costs | Deterministic feeding costs
---|---|---
$\mathbb{E}[\tau^{\text{stoch}}]$ | 1.989 | 2.006
$\mathbb{E}[\tau^{\text{determ}}]$ | 1.993 | 2.005
$V_{0}^{\text{stoch}}$ | 2501648 | 2442247
$V_{0}^{\text{determ}}$ | 2466925 | 2406204
$\mathrm{RI}$ | 1.014 | 1.015

In the scenario with deterministic feeding costs, we see similar results.
Therefore, it seems like the benefit of using a stochastic mortality model is
independent of the chosen feeding cost model since the relative improvements
changed only slightly between the stochastic and deterministic feeding cost
models.

Changing $\mathrm{tc}$ to $1\,\%$ lessens the improvements to roughly
$0.7\,\%$ in the host-parasite model and increases the improvements to
$2.3\,\%$ when changing $\mathrm{tc}$ to $2\,\%$.

##  4 Conclusion and Limitations

In this article, we focused on the question of whether stochastic mortality
models lead to a significant increase in revenue compared to deterministic
mortality models. In the case of a simple host-parasite model, we found an
affirmative answer leading to an increase of $1.5\,\%$ of the farm’s value. In
future studies, we would like to find a more realistic model for the host-
parasite relationship, taking the different phases of the copepodites into
account and therefore using the entire dataset. Also, a biological validation
of the model and its parameters is required to judge how realistic this
mathematical model will be. Another economical validation will be required to
judge how the treatment costs are incorporated into the optimal stopping
problem.

Another idea for future research would involve the consideration of an optimal
treatment strategy instead of just ensuring that the lice threshold is not
exceeded. Maybe preventive mechanical removals in earlier stages could be
beneficial. Moreover, including medical treatments with stochastic delay
differential equations (SDDE) could be of interest.

## Declarations

The authors have no relevant financial or non-financial interests to disclose.

## References

  * [1] Roy M. Anderson and Robert M. May  “Regulation and Stability of Host-Parasite Population Interactions: I. Regulatory Processes”  In _Journal of Animal Ecology_ 47.1 [Wiley, British Ecological Society], 1978, pp. 219–247  DOI: 10.2307/3933
  * [2] Lars Asplin, Jon Albretsen, Ingrid Askeland Johnsen and Anne Dagrun Sandvik  “The hydrodynamic foundation for salmon lice dispersion modeling along the Norwegian coast”  In _Ocean Dynamics_ 70.8, 2020, pp. 1151–1167  DOI: 10.1007/s10236-020-01378-0
  * [3] Sebastian Becker, Patrick Cheridito, Arnulf Jentzen and Timo Welti  “Solving high-dimensional optimal stopping problems using deep learning”  In _European Journal of Applied Mathematics_ 32.3 Cambridge University Press (CUP), 2021, pp. 470–514  DOI: 10.1017/s0956792521000073
  * [4] Christian Oliver Ewald and Kevin Kamm  “On the Impact of Feeding Cost Risk in Aquaculture Valuation and Decision Making”, 2023  arXiv:2309.02970 [q-fin.RM]
  * [5] Christian-Oliver Ewald, Ruolan Ouyang and Tak Kuen Siu  “On the Market-consistent Valuation of Fish Farms: Using the Real Option Approach and Salmon Futures”  In _American Journal of Agricultural Economics_ 99.1, 2017, pp. 207–224  DOI: 10.1093/ajae/aaw052
  * [6] LN Frazer, A Morton and M Krkosek  “Critical thresholds in sea lice epidemics: evidence, sensitivity and subcritical estimation”  In _Proc Biol Sci_ 279, 2012  DOI: 10.1098/rspb.2011.2210
  * [7] Tróndur J. Kragesteen, Knud Simonsen, André W. Visser and Ken H. Andersen  “Optimal salmon lice treatment threshold and tragedy of the commons in salmon farm networks”  In _Aquaculture_ 512, 2019, pp. 734329  DOI: 10.1016/j.aquaculture.2019.734329
  * [8] Manuel Luna, Ignacio Llorente and Ladislao Luna  “A conceptual framework for risk management in aquaculture”  In _Marine Policy_ 147, 2023, pp. 105377  DOI: 10.1016/j.marpol.2022.105377
  * [9] F.A. Milner and C.A. Patton  “A new approach to mathematical modeling of host-parasite systems”  In _Computers & Mathematics with Applications_ 37.2, 1999, pp. 93–110  DOI: 10.1016/S0898-1221(98)00255-7
  * [10] Bard Misund  “Cost Development In Atlantic Salmon and Rainbow Trout Farming: What Is The Cost of Biological Risk?”  In _SSRN_ , 2022  DOI: 10.2139/ssrn.4307278
  * [11] Anne D Sandvik, Ingrid A Johnsen, Mari S Myksvoll, Pål N Sævik and Morten D Skogen  “Prediction of the salmon lice infestation pressure in a Norwegian fjord”  In _ICES Journal of Marine Science_ 77.2, 2020, pp. 746–756  DOI: 10.1093/icesjms/fsz256
  * [12] Christina P. Tadiri, Jude D. Kong, Gregor F. Fussmann, Marilyn E. Scott and Hao Wang  “A Data-Validated Host-Parasite Model for Infectious Disease Outbreaks”  In _Frontiers in Ecology and Evolution_ 7, 2019  DOI: 10.3389/fevo.2019.00307